Real Number System Class 8th Mathematics Term 1 Tamilnadu Board Solution

Class 8^th Mathematics Term 1 Tamilnadu Board Solution

Exercise 1.1

1. The additive identity of rational numbers is ________. Choose the correct…
2. The additive inverse of -3/5 is ______. Choose the correct answer:A. -3/5 B.…
3. The reciprocal of -5/13 is _______. Choose the correct answer:A. 5/13 B. -13/5…
4. The multiplicative inverse of -7 is ___________. Choose the correct answer:A. 7…
5. ________ has no reciprocal. Choose the correct answer:A. 0 B. 1 C. -1 D. 1/4…
6. (-3/7) + 1/9 = 1/9 + (-3/7) Name the property under addition used in each of…
7. 4/9 + (7/8 + 1/2) = (4/9 + 7/8) + 1/2 Name the property under addition used in…
8. 8 + 7/10 = 7/10 + 8 Name the property under addition used in each of the…
9. (-7/15) + 0 = -7/15 = 0 + (-7/15) Name the property under addition used in each…
10. 2/5 + (-2/5) = 0 Name the property under addition used in each of the…
11. 2/3 x 4/5 = 4/5 x 2/3 Name the property under multiplication used in each of…
12. (-3/4) x 1 = -3/4 = 1 x (-3/4) Name the property under multiplication used in…
13. (-17/28) x (-28/17) = 1 Name the property under multiplication used in each of…
14. 1/5 x (7/8 x 4/3) = (1/5 x 7/8) x 4/3 Name the property under multiplication…
15. 2/7 x (9/10 + 2/5) = 2/7 x 9/10 + 2/7 x 2/5 Name the property under…
16. 4 and 2/5 Verify whether commutative property is satisfied for addition,…
17. -3/4 -2/7 Verify whether commutative property is satisfied for addition,…
18. 1/3 , 2/5 -3/7 Verify whether associative property is satisfied for addition,…
19. 2/3 , -4/5 9/10 Verify whether associative property is satisfied for addition,…
20. -5/4 x (8/9 + 5/7) Use distributive property of multiplication of rational…
21. 2/7 x (1/4 - 1/2) Use distributive property of multiplication of rational…

Exercise 1.2

1. 4/3 and 2/5 Find one rational number between the following pairs of rational…
2. -2/7 and 5/6 Find one rational number between the following pairs of rational…
3. 5/11 and 7/8 Find one rational number between the following pairs of rational…
4. 7/4 and 8/3 Find one rational number between the following pairs of rational…
5. 2/7 and 3/5 Find two rational numbers between
6. 6/5 and 9/11 Find two rational numbers between
7. 1/3 and 4/5 Find two rational numbers between
8. -1/6 and 1/3 Find two rational numbers between
9. 1/4 and 1/2 Find three rational numbers between
10. 1/10 and 2/3 Find three rational numbers between
11. -1/3 and 3/2 Find three rational numbers between
12. 1/8 and 1/12 Find three rational numbers between

Exercise 1.3

1. 2 x 5/3 = ________ Choose the correct answer:A. 10/3 B. 2 5/6 C. 10/6 D. 2/3…
2. 2/5 x 4/7 = ________ Choose the correct answer:A. 14/20 B. 8/35 C. 20/14 D.…
3. 2/5 + 4/9 is _________ Choose the correct answer:A. 10/23 B. 8/45 C. 38/45 D.…
4. 1/5 / 2 1/2 is _________ Choose the correct answer:A. 2/25 B. 1/2 C. 10/7 D.…
5. (1 - 1/2) + (3/4 - 1/4) Choose the correct answer:A. 0 B. 1 C. 1/2 D. 3/4…
6. 11/12 / (5/9 x 18/25) Simplify:
7. (2 1/2 x 8/10) / (1 1/2 + 5/8) Simplify:
8. 15/16 (5/6 - 1/2) / 10/11 Simplify:
9. 9/8 / 3/5 (3/4 + 3/5) Simplify:
10. 2/5 / 1/5 [3/4 - 1/2]-1 Simplify:
11. (1 3/4 x 3 1/7) - (4 3/8 / 5 3/5) Simplify:
12. (1/6 + 2 3/4 1 7/11) / 1 1/6 Simplify:
13. (- 1/3) - (2/3 x 1/7) Simplify:

Exercise 1.4

1. am × an is equal to Choose the correct answer for the following:A. am + an B.…
2. p^0 is equal to Choose the correct answer for the following:A. 0 B. 1 C. -1 D.…
3. In 10^2 , the exponent is Choose the correct answer for the following:A. 2 B. 1…
4. 6-1 is equal to Choose the correct answer for the following:A. 6 B. -1 C. - 1/6…
5. The multiplicative inverse of 2-4 is Choose the correct answer for the…
6. (-2)-5 × (-2)^6 is equal to Choose the correct answer for the following:A. -2…
7. (-2)-2 is equal to Choose the correct answer for the following:A. 1/2 B. 1/4 C.…
8. (2^0 + 4-1) × 2^2 is equal to Choose the correct answer for the following:A. 2…
9. (1/3)^-4 is equal to Choose the correct answer for the following:A. 3 B. 3^4 C.…
10. (-1)^50 is equal to Choose the correct answer for the following:A. -1 B. 50 C.…
11. (-4)^5 ÷ (-4)^8 Simplify:
12. (1/2^3)^2 Simplify:
13. (-3)^4 x (5/3)^4 Simplify:
14. (2/3)^5 x (3/4)^2 x (1/5)^2 Simplify:
15. (3-7 ÷ 3^10) × 3-5 Simplify:
16. 2^6 x 3^2 x 2^3 x 3^7/2^8 x 3^6 Simplify:
17. ya - b × yb - c × yc - a Simplify:
18. (4p)^3 × (2p)^2 × p^4 Simplify:
19. Simplify:
20. (1/4)^-2 - 3 x 8^2/3 x x 4^0 + (9/16)^-1/2 Simplify:
21. (3^0 + 4-1) × 2^2 Find the value of:
22. (2-1 × 4-1) ÷ 2-2 Find the value of:
23. (1/2)^-2 + (1/3)^-2 + (1/4)^-2 Find the value of:
24. (3-1 + 4-1 + 5-1)^0 Find the value of:
25. [(-2/3)^-2]^2 Find the value of:
26. 7-20 - 7-21 Find the value of:
27. 5m ÷ 5-3 = 5^5 Find the value of m for which
28. 4m = 64 Find the value of m for which
29. 8m - 3 = 1 Find the value of m for which
30. (a^3)m = a^9 Find the value of m for which
31. (5m)^2 × (25)^3 × 125^2 = 1 Find the value of m for which
32. 2m = (8)^1/3 / (2^3)^2/3 Find the value of m for which
33. If 2x = 16, find i. x ii. 2^x/2 iii. 22x iv. 2x + 2 v. root 2^-x
34. If 3x = 81, find i. x ii. 3x + 3 iii. 3x/2 iv. 32x v. 3x - 6
35. 3^x+1/3^x (x+1) x (3^x/3)^x+1 = 1 Prove that
36. Prove that

Exercise 1.5

1. Just observe the unit digits and state which of the following are not perfect…
2. 78^2 Write down the unit digits of the following:
3. 27^2 Write down the unit digits of the following:
4. 41^2 Write down the unit digits of the following:
5. 35^2 Write down the unit digits of the following:
6. 42^2 Write down the unit digits of the following:
7. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 Find the sum of the following numbers without…
8. 1 + 3 + 5 + 7 Find the sum of the following numbers without actually adding the…
9. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 Find the sum of the following numbers…
10. 7^2 Express the following as a sum of consecutive odd numbers starting with 1…
11. 9^2 Express the following as a sum of consecutive odd numbers starting with 1…
12. 5^2 Express the following as a sum of consecutive odd numbers starting with 1…
13. 11^2 Express the following as a sum of consecutive odd numbers starting with 1…
14. 3/8 Find the squares of the following numbers
15. 7/10 Find the squares of the following numbers
16. 1/5 Find the squares of the following numbers
17. 2/3 Find the squares of the following numbers
18. 31/40 Find the squares of the following numbers
19. (-3)^2 Find the values of the following:
20. (-7)^2 Find the values of the following:
21. (-0.3)^2 Find the values of the following:
22. (- 2/3)^2 Find the values of the following:
23. (- 3/4)^2 Find the values of the following:
24. (-0.6)^2 Find the values of the following:
25. 1^2 + 2^2 + 2^2 = 3^2 , 2^2 + 3^2 + 6^2 = 7^2 3^2 + 4^2 + 12^2 + 13^2 4^2 + 5^2…
26. 11^2 = 121 101^2 = 10201 1001^2 = 1002001 100001^2 = 10000200001 10000001^2 =…

Exercise 1.6

1. 3 × 3 × 4 × 4 Find the square root of each expression given below:…
2. 2 × 2 × 5 × 5 Find the square root of each expression given below:…
3. 3 × 3 × 3 × 3 × 3 × 3 Find the square root of each expression given below:…
4. 5 × 5 × 11 × 11 × 7 × 7 Find the square root of each expression given below:…
5. 9/64 Find the square root of the following:
6. 1/16 Find the square root of the following:
7. 49 Find the square root of the following:
8. 16 Find the square root of the following:
9. 2304 Find the square root of each of the following by Long division method:…
10. 4489 Find the square root of each of the following by Long division method:…
11. 3481 Find the square root of each of the following by Long division method:…
12. 529 Find the square root of each of the following by Long division method:…
13. 3249 Find the square root of each of the following by Long division method:…
14. 1369 Find the square root of each of the following by Long division method:…
15. 5776 Find the square root of each of the following by Long division method:…
16. 7921 Find the square root of each of the following by Long division method:…
17. 576 Find the square root of each of the following by Long division method:…
18. 3136 Find the square root of each of the following by Long division method:…
19. 729 Find the square root of the following numbers by the factorization method:…
20. 400 Find the square root of the following numbers by the factorization method:…
21. 1764 Find the square root of the following numbers by the factorization method:…
22. 4096 Find the square root of the following numbers by the factorization method:…
23. 7744 Find the square root of the following numbers by the factorization method:…
24. 9604 Find the square root of the following numbers by the factorization method:…
25. 5929 Find the square root of the following numbers by the factorization method:…
26. 9216 Find the square root of the following numbers by the factorization method:…
27. 529 Find the square root of the following numbers by the factorization method:…
28. 8100 Find the square root of the following numbers by the factorization method:…
29. 2.56 Find the square root of the following decimal numbers:
30. 7.29 Find the square root of the following decimal numbers:
31. 51.84 Find the square root of the following decimal numbers:
32. 42.25 Find the square root of the following decimal numbers:
33. 31.36 Find the square root of the following decimal numbers:
34. 0.2916 Find the square root of the following decimal numbers:
35. 11.56 Find the square root of the following decimal numbers:
36. 0.001849 Find the square root of the following decimal numbers:
37. 402 Find the least number which must be subtracted from each of the following…
38. 1989 Find the least number which must be subtracted from each of the following…
39. 3250 Find the least number which must be subtracted from each of the following…
40. 825 Find the least number which must be subtracted from each of the following…
41. 4000 Find the least number which must be subtracted from each of the following…
42. 525 Find the least number which must be added to each of the following numbers…
43. 1750 Find the least number which must be added to each of the following numbers…
44. 252 Find the least number which must be added to each of the following numbers…
45. 1825 Find the least number which must be added to each of the following numbers…
46. 6412 Find the least number which must be added to each of the following numbers…
47. 2 Find the square root of the following correct to two places of decimals:…
48. 5 Find the square root of the following correct to two places of decimals:…
49. 0.016 Find the square root of the following correct to two places of decimals:…
50. 7/8 Find the square root of the following correct to two places of decimals:…
51. 1 1/12 Find the square root of the following correct to two places of decimals:…
52. Find the length of the side of a square where area is 441 m^2 .
53. 225/3136 Find the square root of the following:
54. 2116/3481 Find the square root of the following:
55. 529/1764 Find the square root of the following:
56. 7921/5776 Find the square root of the following:

Exercise 1.7

1. Which of the following numbers is a perfect cube? Choose the correct answer for…
2. Which of the following numbers is not a perfect cube? Choose the correct answer…
3. The cube of an odd natural number is Choose the correct answer for the…
4. The number of zeros of the cube root of 1000 is Choose the correct answer for…
5. The unit digit of the cube of the number 50 is Choose the correct answer for…
6. The number of zeros at the end of the cube of 100 is Choose the correct answer…
7. Find the smallest number by which the number 108 must be multiplied to obtain a…
8. Find the smallest number by which the number 88 must be divided to obtain a…
9. The volume of a cube is 64 cm^3 . The side of the cube is Choose the correct…
10. Which of the following is false? Choose the correct answer for the following:A.…
11. 400 Check whether the following are perfect cubes?
12. 216 Check whether the following are perfect cubes?
13. 729 Check whether the following are perfect cubes?
14. 250 Check whether the following are perfect cubes?
15. 1000 Check whether the following are perfect cubes?
16. 900 Check whether the following are perfect cubes?
17. 128 Which of the following numbers are not perfect cubes?
18. 100 Which of the following numbers are not perfect cubes?
19. 64 Which of the following numbers are not perfect cubes?
20. 125 Which of the following numbers are not perfect cubes?
21. 72 Which of the following numbers are not perfect cubes?
22. 625 Which of the following numbers are not perfect cubes?
23. 81 Find the smallest number by which each of the following number must be…
24. 128 Find the smallest number by which each of the following number must be…
25. 135 Find the smallest number by which each of the following number must be…
26. 192 Find the smallest number by which each of the following number must be…
27. 704 Find the smallest number by which each of the following number must be…
28. 625 Find the smallest number by which each of the following number must be…
29. 243 Find the smallest number by which each of the following number must be…
30. 256 Find the smallest number by which each of the following number must be…
31. 72 Find the smallest number by which each of the following number must be…
32. 675 Find the smallest number by which each of the following number must be…
33. 100 Find the smallest number by which each of the following number must be…
34. 729 Find the cube root of each of the following numbers by prime Factorization…
35. 343 Find the cube root of each of the following numbers by prime Factorization…
36. 512 Find the cube root of each of the following numbers by prime Factorization…
37. 0.064 Find the cube root of each of the following numbers by prime…
38. 0.216 Find the cube root of each of the following numbers by prime…
39. 5 23/64 Find the cube root of each of the following numbers by prime…
40. - 1.331 Find the cube root of each of the following numbers by prime…
41. - 27000 Find the cube root of each of the following numbers by prime…
42. The volume of a cubical box is 19.683 cu. cm. Find the length of each side of…

Exercise 1.8

1. 12.568 Express the following correct to two decimal places:
2. 25.416 kg Express the following correct to two decimal places:
3. 39.927 m Express the following correct to two decimal places:
4. 56.596 m Express the following correct to two decimal places:
5. 41.056 m Express the following correct to two decimal places:
6. 729.943 km Express the following correct to two decimal places:
7. 0.0518 m Express the following correct to three decimal places:
8. 3.5327 km Express the following correct to three decimal places:
9. 58.2936l Express the following correct to three decimal places:
10. 0.1327 gm Express the following correct to three decimal places:
11. 365.3006 Express the following correct to three decimal places:
12. 100.1234 Express the following correct to three decimal places:
13. 247 to the nearest ten. Write the approximate value of the following numbers to…
14. 152 to the nearest ten. Write the approximate value of the following numbers to…
15. 6848 to the nearest hundred. Write the approximate value of the following…
16. 14276 to the nearest ten thousand. Write the approximate value of the following…
17. 3576274 to the nearest Lakhs. Write the approximate value of the following…
18. 104, 3567809 to the nearest crore Write the approximate value of the following…
19. 22.266 Round off the following numbers to the nearest integer:
20. 777.43 Round off the following numbers to the nearest integer:
21. 402.06 Round off the following numbers to the nearest integer:
22. 305.85 Round off the following numbers to the nearest integer:
23. 299.77 Round off the following numbers to the nearest integer:
24. 9999.9567 Round off the following numbers to the nearest integer:…

Exercise 1.9

1. 40, 35, 30, ___, ___, ___. Complete the following patterns:
2. 0, 2, 4, ___, ___, ___. Complete the following patterns:
3. 84, 77, 70, ___, ___, ___. Complete the following patterns:
4. 4.4, 5.5, 6.6, ___, ___, ___. Complete the following patterns:
5. 1, 3, 6, 10, ___, ___, ___. Complete the following patterns:
6. 1, 1, 2, 3, 5, 8, 13, 21, ___, ___, ___ (This sequence is called FIBONACCI…
7. 1, 8, 27, 64, ___, ___, ___. Complete the following patterns:
8. A water tank has steps inside it. A monkey is sitting on the top most step. (ie,…
9. A vendor arranged his apples as in the following pattern: A. If there are ten…

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Exercise 1.1

Question 1.

Choose the correct answer:

The additive identity of rational numbers is ________.
A. 0

B. 1

C. -1

D. 2

Answer:

Let, additive identity of rational number is = x

Let, a rational number is = p/q, where q ≠ 0

According to problem,

⇒ p/q + x = p/q

⇒ x = 0

∴ The additive identity of rational number is = 0

Question 2.

Choose the correct answer:

The additive inverse of  is ______.
A. 

B. 

C. 

D. 

Answer:

Let, additive inverse of -3/5 is = x

According to problem,

⇒ -3/5 + x = 0

⇒ x = 3/5

∴ The additive inverse of -3/5 is = 3/5

Question 3.

Choose the correct answer:

The reciprocal of  is _______.
A. 

B. 

C. 

D. 

Answer:

Let, reciprocal of  is = x

According to problem,

∴ The reciprocal of 

Question 4.

Choose the correct answer:

The multiplicative inverse of -7 is ___________.
A. 7

B. 1/7

C. -7

D. -1/7

Answer:

Let, multiplicative inverse of -7 is = x

According to problem,

⇒ (-7) × x = 1

⇒ x = -1/7

∴ The multiplicative inverse of -7 is = -1/7

Question 5.

Choose the correct answer:

________ has no reciprocal.
A. 0

B. 1

C. -1

D. 

Answer:

Among the following numbers 0 has no reciprocal.

∵ reciprocal of 0 = 1/0, which is not possible.

∴ Denominator of a fraction cannot be equal to 0.

Question 6.

Name the property under addition used in each of the following:

Answer:

It is under Commutative property of addition.

Where change in the positions of the operands does not change the result.

Question 7.

Name the property under addition used in each of the following:

Answer:

It is under Associative property of addition.

Where change in the grouping of numbers does not change the result.

Question 8.

Name the property under addition used in each of the following:

Answer:

It is under Commutative property of addition.

Where change in the positions of the operands does not change the result.

Question 9.

Name the property under addition used in each of the following:

Answer:

It is under the property Additive identity.

The sum of 0 and any rational number is the rational number itself.

Question 10.

Name the property under addition used in each of the following:

Answer:

It is under the property additive inverse.

Where addition of two rational numbers is 0.

Question 11.

Name the property under multiplication used in each of the following:

Answer:

It is under Commutative property of multiplication.

Where change in the positions of the operands does not change the result.

Question 12.

Name the property under multiplication used in each of the following:

Answer:

It is under the property multiplicative identity.

The product of any rational number and 1 is the rational number itself.

Question 13.

Name the property under multiplication used in each of the following:

Answer:

It is under the property Multiplicative inverse.

When multiplication of two rational number is = 1.

Question 14.

Name the property under multiplication used in each of the following:

Answer:

It is under Associative property of multiplication.

Where change in the grouping of numbers does not change the result.

Question 15.

Name the property under multiplication used in each of the following:

Answer:

It is under the Distributive property of multiplication over addition.

Where ⇒ a × (b + c) = a × b + a × c

Question 16.

Verify whether commutative property is satisfied for addition, subtraction, multiplication and division of the following pairs of rational numbers.

4 and 

Answer:

A. Commutative property for addition:

We have to prove,

⇒ 4 + 2/5 = 2/5 + 4

LHS,

RHS,

∴ LHS= RHS

∴ Commutative property for addition is satisfied.

B. Commutative property for subtraction:

We have to prove,

⇒ 4 – 2/5 = 2/5 – 4

LHS,

RHS,

∴ LHS ≠ RHS

∴ Commutative property for subtraction is not satisfied.

C. Commutative property for multiplication:

We have to prove,

⇒ 4 × 2/5 = 2/5 × 4

LHS,

RHS,

∴ LHS= RHS

∴ Commutative property for multiplication is satisfied.

D. Commutative property for division:

We have to prove that,

LHS,

⇒ 10

RHS,

∴ LHS ≠ RHS

∴ Commutative property for division is not satisfied.

Question 17.

Verify whether commutative property is satisfied for addition, subtraction, multiplication and division of the following pairs of rational numbers.

Answer:

A. Commutative property for addition:

We have to prove,

LHS,

RHS,

∴ LHS= RHS

∴ Commutative property for addition is satisfied.

B. Commutative property for subtraction:

We have to prove,

LHS,

RHS,

∴ LHS ≠ RHS

∴ Commutative property for subtraction is not satisfied.

C. Commutative property for multiplication:

We have to prove,

LHS,

RHS,

∴ LHS= RHS

∴ Commutative property for multiplication is satisfied.

D. Commutative property for division:

We have to prove that,

LHS,

⇒ 21/8

RHS,

∴ LHS ≠ RHS

∴ Commutative property for division is not satisfied.

Question 18.

Verify whether associative property is satisfied for addition, subtraction, multiplication and division of the following pairs of rational numbers.

Answer:

A. Associative property for addition:

We have to prove that,

LHS,

RHS,

∴ LHS = RHS

∴ Associative property for addition is satisfied.

B. Associative property for Subtraction:

We have to prove that,

LHS,

RHS,

∴ LHS ≠ RHS

∴ Associative property for Subtraction is not satisfied.

C. Associative property for multiplication:

We have to prove that,

LHS,

RHS,

∴ LHS= RHS

∴ Associative property for multiplication is satisfied.

D. Associative Property for division:

We have to prove that,

LHS,

RHS,

∴ LHS ≠ RHS

∴ Associative property for Division is not satisfied.

Question 19.

Verify whether associative property is satisfied for addition, subtraction, multiplication and division of the following pairs of rational numbers.

Answer:

A. Associative property for addition:

We have to prove that,

LHS,

RHS,

∴ LHS = RHS

∴ Associative property for addition is satisfied.

B. Associative property for Subtraction:

We have to prove that,

LHS,

RHS,

∴ LHS ≠ RHS

∴ Associative property for Subtraction is not satisfied.

C. Associative property for multiplication:

We have to prove that,

LHS,

RHS,

∴ LHS= RHS

∴ Associative property for multiplication is satisfied.

D. Associative Property for division:

We have to prove that,

LHS,

RHS,

∴ LHS ≠ RHS

∴ Associative property for Divission is not satisfied.

Question 20.

Use distributive property of multiplication of rational numbers and simplify:

Answer:

According to distributive property of multiplication of rational numbers,

Question 21.

Use distributive property of multiplication of rational numbers and simplify:

Answer:

According to distributive property of multiplication of rational numbers,

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Exercise 1.2

Question 1.

Find one rational number between the following pairs of rational numbers.

 and 

Answer:

Let, q is the rational number between 4/3 and 2/5.

∴ one rational number between 4/3 and 2/5 is = 13/15

Question 2.

Find one rational number between the following pairs of rational numbers.

 and 

Answer:

Let, q is the rational number between -2/7 and 5/6

∴ one rational number between -2/7 and 5/6 is = 23/84

Question 3.

Find one rational number between the following pairs of rational numbers.

 and 

Answer:

Let, q is the rational number between 5/11 and 7/8

∴ one rational number between 5/11 and 7/8 is = 117/176

Question 4.

Find one rational number between the following pairs of rational numbers.

 and 

Answer:

Let, q is the rational number between 7/4 and 8/3

∴ one rational number between 7/4 and 8/3 is = 53/24

Question 5.

Find two rational numbers between

 and 

Answer:

Let, p and q are two rational numbers between 2/7 and 3/5

⇒ p = 31/70

⇒ q = 51/140

∴ Two rational numbers between 2/7 and 3/5 is = 31/70 and 51/140

Question 6.

Find two rational numbers between

 and 

Answer:

Let, p and q are two rational numbers between 6/5 and 9/11

∴ Two rational numbers between 6/5 and 9/11 is = 111/110 and 243/220

Question 7.

Find two rational numbers between

 and 

Answer:

Let, p and q are two rational numbers between 1/3 and 4/5

∴ Two rational numbers between 1/3 amd 4/5 is = 17/30 and 9/20

Question 8.

Find two rational numbers between

 and 

Answer:

Let, p and q are two rational numbers between -1/6 and 1/3

∴ Two rational numbers between -1/6 and 1/3 is = 1/12 and -1/24

Question 9.

Find three rational numbers between

 and 

Answer:

Let, p, q and r are three rational numbers between 1/4 and 1/2

∴ Three rational numbers between 1/4 and 1/2 is = 3/8, 5/16 and 9/32

Question 10.

Find three rational numbers between

 and 

Answer:

Let, p, q and r are three rational numbers between 1/10 and 2/3

∴ Three rational numbers between 1/10 and 2/3 is = 23/60, 29/120, 41/240

Question 11.

Find three rational numbers between

 and 

Answer:

Let, p, q and r are three rational numbers between -1/3 and 3/2

∴ Three rational numbers between -1/3 and 3/2 is = 7/12, 1/8 and -5/48

Question 12.

Find three rational numbers between

 and 

Answer:

Let, p, q and r are three rational numbers between 1/8 and 1/12

∴ Three rational numbers between 1/8 and 1/12 is = 5/48, 11/96 and 23/192

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Exercise 1.3

Question 1.

Choose the correct answer:

 = ________
A. 

B. 

C. 

D. 

Answer:

(B) doesn’t match the solution.

(C) doesn’t match the solution.

(D) doesn’t match the solution.

Question 2.

Choose the correct answer:

 = ________
A. 

B. 

C. 

D. 

Answer:

(A) doesn’t match the solution.

(C) doesn’t match the solution.

(D) doesn’t match the solution.

Question 3.

Choose the correct answer:

 is _________
A. 

B. 

C. 

D. 

Answer:

(A) doesn’t match the solution.

(B) doesn’t match the solution.

(D) doesn’t match the solution.

Question 4.

Choose the correct answer:

 is _________
A. 

B. 

C. 

D. 

Answer:

(B) doesn’t match the solution.

(C) doesn’t match the solution.

(D) doesn’t match the solution.

Question 5.

Choose the correct answer:


A. 0

B. 1

C. 

D. 

Answer:

where LCM is least common multiple)

, where LCM is least common multiple)

(A) doesn’t match the solution.

(C) doesn’t match the solution.

(D) doesn’t match the solution.

Question 6.

Simplify:

Answer:

Question 7.

Simplify:

Answer:

Question 8.

Simplify:

Answer:

where LCM is least common multiple

Question 9.

Simplify:

Answer:

Question 10.

Simplify:

Answer:

Question 11.

Simplify:

Answer:

Question 12.

Simplify:

Answer:

Question 13.

Simplify:

Answer:

---

Exercise 1.4

Question 1.

Choose the correct answer for the following:

a^m × aⁿ is equal to
A. a^m + aⁿ

B. a^{m – n}

C. a^{m + n}

D. a^mn

Answer:

a^m×aⁿ = (a×a×a×a×……m times)×(a×a×a×a×……n times)

= a× a× a× a× ……m + n times

⇒ a^m × aⁿ = a^{m + n}

(Also, Product Rule: a^m × aⁿ = a^{m + n} , where ‘a’ is a real no. and m, n are positive integers.)

(A) Doesn’t match the solution.

(B) Doesn’t match the solution.

(D) Doesn’t match the solution.

Question 2.

Choose the correct answer for the following:

p⁰ is equal to
A. 0

B. 1

C. -1

D. p

Answer:

For p≠0,

p⁰ = pⁿ ÷ pⁿ

⇒ p⁰ = 1

(Also, Number with zero exponent rule:If ‘a’ is a rational no. other than zero, then a⁰ = 1)

(A) Doesn’t match the solution.

(C) Doesn’t match the solution.

(D) Doesn’t match the solution.

Question 3.

Choose the correct answer for the following:

In 10², the exponent is
A. 2

B. 1

C. 10

D. 100

Answer:(D)

Answer:

10² = 10× 10 (∵ aⁿ = a× a× a× …… n times, where n is a positive integer)

⇒ 10² = 100

Question 4.

Choose the correct answer for the following:

6^-1 is equal to
A. 6

B. -1

C. 

D. 

Answer:(D)

Answer:

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

(A) doesn’t match the solution.

(B) doesn’t match the solution.

(C) doesn’t match the solution.

Question 5.

Choose the correct answer for the following:

The multiplicative inverse of 2^-4 is
A. 2

B. 4

C. 2⁴

D. -4

Answer:(C)

Answer:

Let a be the multiplicative inverse of 2^-4

⇒ 2^-4 × a = 1 (by definition of multiplicative inverse)

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

⇒ a = 2⁴

(A) doesn’t match the solution.

(B) doesn’t match the solution.

(D) doesn’t match the solution.

Question 6.

Choose the correct answer for the following:

(-2)^-5 × (-2)⁶ is equal to
A. -2

B. 2

C. -5

D. 6

Answer:

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

( Product Rule: a^m × aⁿ = a^{m + n} , where ‘a’ is a real no. and m, n are positive integers.)

⇒ (-2)^-5 × (-2)⁶ = -2

(B) doesn’t match the solution.

(C) doesn’t match the solution.

(D) doesn’t match the solution.

Question 7.

Choose the correct answer for the following:

(-2)^-2 is equal to
A. 

B. 

C. 

D. 

Answer:

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

(A) doesn’t match the solution.

(C) doesn’t match the solution.

(D) doesn’t match the solution.

Question 8.

Choose the correct answer for the following:

(2⁰ + 4^-1) × 2² is equal to
A. 2

B. 5

C. 4

D. 3

Answer:

(2⁰ + 4^-1) × 2² = (1 + 4^-1) × 2²

(∵ Number with zero exponent rule:If ‘a’ is a rational no. other than zero, then a⁰ = 1)

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

(A) doesn’t match the solution.

(C) doesn’t match the solution.

(D) doesn’t match the solution.

Question 9.

Choose the correct answer for the following:

 is equal to
A. 3

B. 3⁴

C. 1

D. 3^-4

Answer:

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

(A) doesn’t match the solution.

(C) doesn’t match the solution.

(D) doesn’t match the solution.

Question 10.

Choose the correct answer for the following:

(-1)⁵⁰ is equal to
A. -1

B. 50

C. -50

D. 1

Answer:(D)

Answer:

(-1)⁵⁰ = -1×-1×-1×-1………50 times

⇒ (-1)⁵⁰ = 1 (∵ 50 is even no.)

(∵ (-1)ⁿ = -1, if n is odd and (-1)ⁿ = 1, if n is even)

(A) doesn’t match the solution.

(B) doesn’t match the solution.

(C) doesn’t match the solution.

Question 11.

Simplify:

(-4)⁵ ÷ (-4)⁸

Answer:

⇒ (-4)⁵ ÷ (-4)⁸ = (-4)^5-8

where ‘a’ is a non-zero real no. and m, n are positive integers.)

⇒ (-4)⁵ ÷ (-4)⁸ = (-4)^-3

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

Question 12.

Simplify:

Answer:

where b≠0, a and b are real numbers, m is an integer)

Question 13.

Simplify:

Answer:

where b≠0, a and b are real numbers, m is an integer)

Question 14.

Simplify:

Answer:

where b≠0, a and b are real numbers, m is an integer)

where ‘a’ is a non-zero real no. and m, n are positive integers.)

Question 15.

Simplify:

(3^-7 ÷ 3¹⁰) × 3^-5

Answer:

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

(Product Rule: a^m × aⁿ = a^{m + n} , where ‘a’ is a real no. and m, n are positive integers.)

Question 16.

Simplify:

Answer:

(Product Rule: a^m × aⁿ = a^{m + n} , where ‘a’ is a real no. and m, n are positive integers.)

where ‘a’ is a non-zero real no. and m, n are positive integers.)

Question 17.

Simplify:

y^{a – b} × y^{b – c} × y^{c - a}

Answer:

y^{a – b} × y^{b – c} × y^{c – a} = y^{(a-b + b-c + c-a)}

(Product Rule: a^m × aⁿ = a^{m + n} , where ‘a’ is a real no. and m, n are positive integers.)

⇒ y^{a – b} × y^{b – c} × y^{c – a} = y^{(a-a + b-b-c + c)}

⇒ y^{a – b} × y^{b – c} × y^{c – a} = y⁰

⇒ y^{a – b} × y^{b – c} × y^{c – a} = 1

(∵ Number with zero exponent rule: If ‘a’ is a rational no. other than zero, then a⁰ = 1)

Question 18.

Simplify:

(4p)³ × (2p)² × p⁴

Answer:

(4p)³ × (2p)² × p⁴ = 4³× p³× 2²× p²× p⁴

⇒ (4p)³ × (2p)² × p⁴ = 4×4× 4× p³× 2×2× p²× p⁴

⇒ (4p)³ × (2p)² × p⁴ = 256 p^{3 + 2 + 4}

(Product Rule: a^m × aⁿ = a^{m + n} , where ‘a’ is a real no. and m, n are positive integers.)

⇒ (4p)³ × (2p)² × p⁴ = 256 p⁹

Question 19.

Simplify:

Answer:

(∵ Number with zero exponent rule: If ‘a’ is a rational no. other than zero, then a⁰ = 1)

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

(Product Rule: a^m × aⁿ = a^{m + n} , where ‘a’ is a real no. and m, n are positive integers.)

Now, taking 9^1/2 common, we get-

Question 20.

Simplify:

Answer:

(∵ Number with zero exponent rule: If ‘a’ is a rational no. other than zero, then a⁰ = 1)

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

where b≠0, a and b are real numbers, m is an integer)

where ‘a’ is a non-zero real no. and m, n are positive integers.)

Question 21.

Find the value of:

(3⁰ + 4^-1) × 2²

Answer:

(3⁰ + 4^-1) × 2² = (1 + 4^-1) × 2²

(∵ Number with zero exponent rule: If ‘a’ is a rational no. other than zero, then a⁰ = 1)

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

⇒3⁰ + 4^-1× 2² = 5

Question 22.

Find the value of:

(2^-1 × 4^-1) ÷ 2^-2

Answer:

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

we can write it as,

Answer.

Question 23.

Find the value of:

Answer:

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

Question 24.

Find the value of:
(3^-1 + 4^-1 + 5^-1)⁰

Answer:

Consider(3^-1 + 4^-1 + 5^-1)⁰,
Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then 
So,

Take the L.C.M of 3, 4 and 5 = 60

Number with zero exponent rule: If ‘a’ is a rational no. other than zero, then a⁰ = 1

Question 25.

Find the value of:

Answer:

(∵ if ‘a’ is a real no. and m, n are integers, then (a^m)ⁿ = a^mn)

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

Question 26.

Find the value of:

7^-20 – 7^-21

Answer:

Question 27.

Find the value of m for which

5^m ÷ 5^-3 = 5⁵

Answer:

5^m ÷ 5^-3 = 5⁵

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

where ‘a’ is a non-zero real no. and m, n are positive integers.)

⇒ m = 2 (∵ base is same)

Question 28.

Find the value of m for which

4^m = 64

Answer:

4^m = 64

∵ 4³ = 64

⇒ 4^m = 4³

⇒ m = 3 (∵ base is same)

Question 29.

Find the value of m for which

8^{m – 3} = 1

Answer:

8^{m – 3} = 1

where ‘a’ is a non-zero real no. and m, n are positive integers.)

⇒ 8^m = 1× 8³

⇒ 8^m = 8³

⇒ m = 3 (∵ base is same)

Question 30.

Find the value of m for which

(a³)^m = a⁹

Answer:

(a³)^m = a⁹

⇒ a^3m = a⁹

(∵ if ‘a’ is a real no. and m, n are integers, then (a^m)ⁿ = a^mn)

⇒ 3m = 9 (∵ base is same)

⇒ m = 3

Question 31.

Find the value of m for which

(5^m)² × (25)³ × 125² = 1

Answer:

(5^m)² × (25)³ × 125² = 1

⇒ (5²)^m × (25)³ × 125² = 1

⇒ 5^2m × (5²)³ × (5³)² = 1

(∵ 125 = 5× 5× 5 = 5³, and 25 = 5× 5 = 5²)

⇒ (5)^2m × (5)⁶ × (5)⁶ = 1

(∵ if ‘a’ is a real no. and m, n are integers, then (a^m)ⁿ = a^mn)

⇒ 5^{2m + 3 + 6} = 25⁰ (∵ 25⁰ = 1)

⇒ 2m + 6 + 6 = 0 (∵ base is same)

⇒ 2m + 12 = 0

⇒ 2m = -12

⇒ m = -6

Question 32.

Find the value of m for which

Answer:

Question 33.

If 2^x = 16, find

i. x

ii. 

iii. 2^2x

iv. 2^{x + 2}

v. 

Answer:

i. 2^x = 16

⇒ 2^x = 2⁴

⇒ x = 4 (∵ base is same)

ii. 

iii. 2^2x = 2^{2× 4}

⇒ 2^2x = 2⁸

⇒ 2^2x = (2× 2× ……8 times)

⇒ 2^2x = 256

iv. 2^{x + 2} = 2^{4 + 2}

⇒ 2^{x + 2} = 2⁶

⇒ 2^{x + 2} = 2× 2× 2× 2× 2× 2

⇒ 2^{x + 2} = 64

v. √2^-x = √2^-4

Question 34.

If 3^x = 81, find

i. x

ii. 3^{x + 3}

iii. 3^x/2

iv. 3^2x

v. 3^{x – 6}

Answer:

i. 3^x = 81

⇒ 3^x = 3⁴ (∵ 3⁴ = 81)

⇒ x = 4 (∵ base is same)

ii. 3^{x + 3} = 3^{4 + 3}

⇒ 3^{x + 3} = 3⁷

⇒ 3^{x + 3} = 3× 3× 3× 3× 3× 3× 3

⇒ 3^{x + 3} = 2187

iii. 

iv. 3^2x = 3^2×4

⇒ 3^2x = 3⁸

⇒ 3^2x = 3 × 3× 3× 3× 3× 3× 3× 3

⇒ 3^2x = 6561

v. 3^{x – 6} = 3^{4 – 6}

⇒ 3^{x – 6} = 3^–2

(∵ Reciprocal law: If ‘a’ is a real no. and m is a positive integer, then  )

Question 35.

Prove that

Answer:

Taking L.H.S,

(∵ if ‘a’ is a real no. and m, n are integers, then (a^m)ⁿ = a^mn)

where b≠0, a and b are real numbers, m is an integer)

(By Power of Product Rule)

Hence, proved.

Question 36.

Prove that

Answer:

Taking L.H.S,

where b≠0, a and b are real numbers, m is an integer)

(∵ if ‘a’ is a real no. and m, n are integers, then (a^m)ⁿ = a^mn)

( Product Rule: a^m × aⁿ = a^{m + n} , where ‘a’ is a real no. and m, n are positive integers.)

where ‘a’ is a non-zero real no. and m, n are positive integers.)

(∵ Number with zero exponent rule: If ‘a’ is a rational no. other than zero, then a⁰ = 1)

Hence, proved.

---

Exercise 1.5

Question 1.

Just observe the unit digits and state which of the following are not perfect squares.

i. 3136

ii. 3722

iii. 9348

iv. 2304

v. 8343

Answer:

We know that perfect squares end with the digits 0, 1, 4, 6, 9. Anything other than these in the unit’s place does not qualify to be a perfect square.

Therefore (ii) 3722 is not a perfect square

(iii) 9348 is not a perfect square

(v) 8343 is not a perfect square

Question 2.

Write down the unit digits of the following:

78²

Answer:

78 × 78

Let us only consider the unit’s digits of 78, which is 8.

8 × 8 = 64, which has 4 in its unit’s place.

This number is retained in the unit’s place when the digit is squared.

Hence the answer is 4.

Question 3.

Write down the unit digits of the following:

27²

Answer:

27 × 27

Let us only consider the unit’s digits of 27, which is 7.

7 × 7 = 49, which has 9 in its unit’s place.

This number is retained in the unit’s place when the digit is squared.

Hence the answer is 9.

Question 4.

Write down the unit digits of the following:

41²

Answer:

41 × 41

Let us only consider the unit’s digits of 41, which is 1.

1 × 1 = 1, which has 1 in its unit’s place.

This number is retained in the unit’s place when the digit is squared.

Hence the answer is 1.

Question 5.

Write down the unit digits of the following:

35²

Answer:

35 × 35

Let us only consider the unit’s digits of 35, which is 5.

5 × 5 = 25, which has 5 in its unit’s place.

This number is retained in the unit’s place when the digit is squared.

Hence the answer is 5.

Question 6.

Write down the unit digits of the following:

42²

Answer:

42 × 42

Let us only consider the unit’s digits of 42, which is 2.

2 × 2 = 4, which has 4 in its unit’s place.

This number is retained in the unit’s place when the digit is squared.

Hence the answer is 4.

Question 7.

Find the sum of the following numbers without actually adding the numbers.

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15

Answer:

We know that the sum of first n odd numbers is n²

Since there are 8 odd numbers starting from 1 to 15,

Sum = 8²

= 64

Question 8.

Find the sum of the following numbers without actually adding the numbers.

1 + 3 + 5 + 7

Answer:

We know that the sum of first n odd numbers is n²

Since there are 4 odd numbers starting from 1 to 7,

Sum = 4²

= 16

Question 9.

Find the sum of the following numbers without actually adding the numbers.

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17

Answer:

We know that the sum of first n odd numbers is n²

Since there are 9 odd numbers starting from 1 to 17,

Sum = 9²

= 81

Question 10.

Express the following as a sum of consecutive odd numbers starting with 1

7²

Answer:

We know that any square can be expressed as the sum of consecutive odd numbers starting from 1.

n² = 1 + 3 + 5 + … + (2n-1)

Therefore, 7² = 1 + 3 + 5 + … + (2 × 7-1)

= 1 + 3 + 5 + … + 13

= 1 + 3 + 5 + 7 + 9 + 11 + 13

Question 11.

Express the following as a sum of consecutive odd numbers starting with 1

9²

Answer:

We know that any square can be expressed as the sum of consecutive odd numbers starting from 1.

n² = 1 + 3 + 5 + … + (2n-1)

Therefore, 7² = 1 + 3 + 5 + … + (2 × 9-1)

= 1 + 3 + 5 + … + 17

= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17

Question 12.

Express the following as a sum of consecutive odd numbers starting with 1

5²

Answer:

We know that any square can be expressed as the sum of consecutive odd numbers starting from 1.

n² = 1 + 3 + 5 + … + (2n-1)

Therefore, 7² = 1 + 3 + 5 + … + (2 × 5-1)

= 1 + 3 + 5 + … + 9

= 1 + 3 + 5 + 7 + 9

Question 13.

Express the following as a sum of consecutive odd numbers starting with 1

11²

Answer:

We know that any square can be expressed as the sum of consecutive odd numbers starting from 1.

n² = 1 + 3 + 5 + … + (2n-1)

Therefore, 11² = 1 + 3 + 5 + … + (2 × 11-1)

= 1 + 3 + 5 + … + 21

= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

Question 14.

Find the squares of the following numbers

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 15.

Find the squares of the following numbers

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 16.

Find the squares of the following numbers

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 17.

Find the squares of the following numbers

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 18.

Find the squares of the following numbers

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 19.

Find the values of the following:

(-3)²

Answer:

(-3)² can be written as (-3) × (-3)

= + (3 × 3) = 9 (negative × negative = positive)

Question 20.

Find the values of the following:

(-7)²

Answer:

(-7)² can be written as (-7) × (-7)

= + (7 × 7) = 49 (negative × negative = positive)

Question 21.

Find the values of the following:

(-0.3)²

Answer:

(-0.3)² can be written as (-0.3) × (-0.3)

= + (0.3 × 0.3) = 0.09 (negative × negative = positive)

Question 22.

Find the values of the following:

Answer:

(-)² can be written as (-) × (-)

= + () = (negative × negative = positive)

Question 23.

Find the values of the following:

Answer:

(-)² can be written as (-) × (-)

= + () = (negative × negative = positive)

Question 24.

Find the values of the following:

(-0.6)²

Answer:

(-0.6)² can be written as (-0.6) × (-0.6)

= + (0.6 × 0.6) = 0.36 (negative × negative = positive)

Question 25.

Using the given pattern, find the missing numbers:

1² + 2² + 2² = 3²,

2² + 3² + 6² = 7²

3² + 4² + 12² + 13²

4² + 5² + _____ = 21²

5² + ____ + 30² = 31²

6² + 7² + ____ = _____

Answer:

In every line, the third number (without any power) is the product of first two numbers.

i.e, 1 × 2 = 2, 2 × 3 = 6 and so on.

Using the same logic,

4 × 5 = 20, therefore the missing number is 20²

5 × y = 30, therefore y = 6 and the missing number is 6²

6 × 7 = 42, therefore the missing number is 42²

We also see that the right-hand side of every equation is one more than the last number on the left hand side (without any power)

So 6² + 7² + 42² = 43²

Question 26.

Using the given pattern, find the missing numbers:

11² = 121

101² = 10201

1001² = 1002001

100001² = 10000200001

10000001² = 100000020000001

Answer:

The number of zeroes in the answer is twice the number of zeroes in the base of the square, separated by the digit 2.

---

Exercise 1.6

Question 1.

Find the square root of each expression given below:

3 × 3 × 4 × 4

Answer:

We have 

=

= (Property of roots)

= 3 × 4

= 12

Question 2.

Find the square root of each expression given below:

2 × 2 × 5 × 5

Answer:

We have 

=

= (Property of roots)

= 2 × 5

= 10

Question 3.

Find the square root of each expression given below:

3 × 3 × 3 × 3 × 3 × 3

Answer:

We have 

=

= (Property of roots)

= 3 × 3 × 3

= 27

Question 4.

Find the square root of each expression given below:

5 × 5 × 11 × 11 × 7 × 7

Answer:

We have 

=

= (Property of roots)

= 5 × 11 × 7

= 385

Question 5.

Find the square root of the following:

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 6.

Find the square root of the following:

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 7.

Find the square root of the following:

49

Answer:

=

= 7

Question 8.

Find the square root of the following:

16

Answer:

=

= 4

Question 9.

Find the square root of each of the following by Long division method:

2304

Answer:

∴48

Question 10.

Find the square root of each of the following by Long division method:

4489

Answer:

∴67

Question 11.

Find the square root of each of the following by Long division method:

3481

Answer:

∴59

Question 12.

Find the square root of each of the following by Long division method:

529

Answer:

∴23

Question 13.

Find the square root of each of the following by Long division method:

3249

Answer:

∴57

Question 14.

Find the square root of each of the following by Long division method:

1369

Answer:

∴37

Question 15.

Find the square root of each of the following by Long division method:

5776

Answer:

∴76

Question 16.

Find the square root of each of the following by Long division method:

7921

Answer:

∴89

Question 17.

Find the square root of each of the following by Long division method:

576

Answer:

∴24

Question 18.

Find the square root of each of the following by Long division method:

3136

Answer:

∴56

Question 19.

Find the square root of the following numbers by the factorization method:

729

Answer:

∴ 729 = 3 × 3 × 3 × 3 × 3 × 3

So, = 3 × 3 × 3 = 27

Question 20.

Find the square root of the following numbers by the factorization method:

400

Answer:

∴ 400 = 2 × 2 × 2 × 2 × 5 × 5

So, = 2 × 2 × 5 = 20

Question 21.

Find the square root of the following numbers by the factorization method:

1764

Answer:

∴ 1764 = 2 × 2 × 3 × 3 × 7 × 7

So, = 2 × 3 × 7 = 42

Question 22.

Find the square root of the following numbers by the factorization method:

4096

Answer:

∴ 4096 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

So, = 2 × 2 × 2 × 2 × 2 × 2 = 64

Question 23.

Find the square root of the following numbers by the factorization method:

7744

Answer:

∴ 7744 = 2 × 2 × 2 × 2 × 2 × 2 × 11 × 11

So, = 2 × 2 × 2 × 11 = 88

Question 24.

Find the square root of the following numbers by the factorization method:

9604

Answer:

∴ 9604 = 2 × 2 × 7 × 7 × 7 × 7

So, = 2 × 7 × 7 = 98

Question 25.

Find the square root of the following numbers by the factorization method:

5929

Answer:

∴ 5929 = 7 × 7 × 11 × 11

So, = 7 × 11 = 77

Question 26.

Find the square root of the following numbers by the factorization method:

9216

Answer:

∴ 9216 = 2 × 2 × 2 × 2 × 2 × 2 × 12 × 12

So, = 2 × 2 × 2 × 12 = 96

Question 27.

Find the square root of the following numbers by the factorization method:

529

Answer:

∴ 529 = 23 × 23

So, = 23

Question 28.

Find the square root of the following numbers by the factorization method:

8100

Answer:

∴ 8100 = 10 × 10 × 9 × 9

So, = 10 × 9 = 90

Question 29.

Find the square root of the following decimal numbers:

2.56

Answer:

∴1.6

Question 30.

Find the square root of the following decimal numbers:

7.29

Answer:

∴2.7

Question 31.

Find the square root of the following decimal numbers:

51.84

Answer:

∴7.2

Question 32.

Find the square root of the following decimal numbers:

42.25

Answer:

∴6.5

Question 33.

Find the square root of the following decimal numbers:

31.36

Answer:

∴5.6

Question 34.

Find the square root of the following decimal numbers:

0.2916

Answer:

∴0.54

Question 35.

Find the square root of the following decimal numbers:

11.56

Answer:

∴3.4

Question 36.

Find the square root of the following decimal numbers:

0.001849

Answer:

∴0.043

Question 37.

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square:

402

Answer:

402-2 = 400 = 20²

∴ 2 must be subtracted from 402 to get a perfect square.

Question 38.

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square:

1989

Answer:

1989-53 = 1936 = 44²

∴ 53 must be subtracted from 1989 to get a perfect square.

Question 39.

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square:

3250

Answer:

3250-1 = 3249 = 57²

∴ 1 must be subtracted from 3250 to get a perfect square.

Question 40.

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square:

825

Answer:

825-49 = 776 = 26²

∴ 49 must be subtracted from 825 to get a perfect square.

Question 41.

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square:

4000

Answer:

4000-31 = 3969 = 63²

∴ 31 must be subtracted from 4000 to get a perfect square.

Question 42.

Find the least number which must be added to each of the following numbers so as to get a perfect square:

525

Answer:

525 + 4 = 529 = 23²

∴ 4 must be added to 525 to get a perfect square.

Question 43.

Find the least number which must be added to each of the following numbers so as to get a perfect square:

1750

Answer:

1750 + 14 = 1764 = 42²

∴ 14 must be added to 1750 to get a perfect square.

Question 44.

Find the least number which must be added to each of the following numbers so as to get a perfect square:

252

Answer:

252 + 4 = 256 = 16²

∴ 4 must be added to 252 to get a perfect square.

Question 45.

Find the least number which must be added to each of the following numbers so as to get a perfect square:

1825

Answer:

1825 + 24 = 1849 = 43²

∴ 24 must be added to 1825 to get a perfect square.

Question 46.

Find the least number which must be added to each of the following numbers so as to get a perfect square:

6412

Answer:

6412 + 149 = 6561 = 81²

∴ 149 must be added to 6412 to get a perfect square.

Question 47.

Find the square root of the following correct to two places of decimals:

2

Answer:

∴1.41

Question 48.

Find the square root of the following correct to two places of decimals:

5

Answer:

∴2.23

Question 49.

Find the square root of the following correct to two places of decimals:

0.016

Answer:

∴0.12

Question 50.

Find the square root of the following correct to two places of decimals:

Answer:

∴0.93

Question 51.

Find the square root of the following correct to two places of decimals:

Answer:

∴1.04

Question 52.

Find the length of the side of a square where area is 441 m².

Answer:

Let the length of the side be x m.

So Area = x × x = x² = 441 m²

x =

= 21 m

Question 53.

Find the square root of the following:

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 54.

Find the square root of the following:

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 55.

Find the square root of the following:

Answer:

We know that  for all real values of x and y.

Therefore, 

=

Question 56.

Find the square root of the following:

Answer:

We know that  for all real values of x and y.

Therefore, 

=

---

Exercise 1.7

Question 1.

Choose the correct answer for the following:

Which of the following numbers is a perfect cube?
A. 125

B. 36

C. 75

D. 100

Answer:

A. 125

Prime factorization of 125:

⇒ 125 = 5 × 5 × 5 = 5³

∴ 125 is a perfect cube.

B. 36

Prime factorization:

⇒ 36 = 2 × 2 × 3 × 3

There are only two 2’s and two 3’s.

∴ 36 is not a perfect cube.

C. 75

Prime Factorization:

⇒ 75 = 5 × 5 × 3

There are only two 5’s and one 3.

∴ 75 is not a perfect cube.

D. 100

Prime Factorization:

⇒ 100 = 2 × 2 × 5 × 5

There are only two 2’s and two 5’s.

Hence, 100 is not a perfect cube.

Question 2.

Choose the correct answer for the following:

Which of the following numbers is not a perfect cube?
A. 1331

B. 512

C. 343

D. 100

Answer:

A. 1331

Prime Factorization:

⇒ 1331 = 11 × 11 × 11 = 11³

∴ 1331 is a perfect cube.

B. 512

Prime Factorization:

⇒ 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³ × 2³

= 8³

∴ 512 is a perfect cube.

C. 343

Prime Factorization:

⇒ 343 = 7 × 7 × 7 = 7³

∴ 343 is a perfect cube.

D. 100

Prime Factorization:

⇒ 100 = 2 × 2 × 5 × 5

There are only two 2’s and two 5’s.

Hence, 100 is not a perfect cube.

Question 3.

Choose the correct answer for the following:

The cube of an odd natural number is
A. Even

B. Odd

C. May be even, May be odd

D. Prime number

Answer:

We know that cubes of odd number are all odds.

Question 4.

Choose the correct answer for the following:

The number of zeros of the cube root of 1000 is
A. 1

B. 2

C. 3

D. 4

Answer:

Prime Factorization:

⇒  = (1000)^1/3

(1000)^1/3= ((2 × 2 × 2) × (5 × 5 × 5))^1/3

= (2³ × 5³)^1/3

= (10³)^1/3

We know that by law of exponents, (a^m)ⁿ = a^mn.

∴ (1000)^1/3= 10

Hence, there is only one zero in the cube root of 1000.

Question 5.

Choose the correct answer for the following:

The unit digit of the cube of the number 50 is
A. 1

B. 0

C. 5

D. 4

Answer:

We know that the cubes of the numbers with 0 as unit digit will have the same unit digit i.e. 0.

∴ The unit digit of the cube of the number 50 is 0.

Question 6.

Choose the correct answer for the following:

The number of zeros at the end of the cube of 100 is
A. 1

B. 2

C. 4

D. 6

Answer:

Cube of 100 = 100³

= 100 × 100 × 100

= 1000000

∴ There are 6 zeros at the end of the cube of 100.

Question 7.

Choose the correct answer for the following:

Find the smallest number by which the number 108 must be multiplied to obtain a perfect cube
A. 2

B. 3

C. 4

D. 5

Answer:

Prime Factorization:

⇒ 108 = 2 × 2 × 3 × 3 × 3

In the above Factorization, 2 × 2 remains after grouping the 3’s in triplets.

∴ 108 is not a perfect cube.

To make it a perfect cube, we multiply it by 2.

Prime Factorization:

⇒ 108 × 2 = 2 × 2 × 2 × 3 × 3 × 3

⇒ 216 = 2³ × 3³

= (2 × 3)³

= 6³ which is a perfect cube.

∴ The smallest number by which the number 108 must be multiplied to obtain a perfect cube is 2.

Question 8.

Choose the correct answer for the following:

Find the smallest number by which the number 88 must be divided to obtain a perfect cube
A. 11

B. 5

C. 7

D. 9

Answer:

Prime Factorization:

⇒ 88 = 2 × 2 × 2 × 11

The prime factor 11 does not appear in triplet.

∴ 88 is not a perfect cube.

Since in factorization, 11 appear only one time, we should divide the number 80 by 11.

⇒ 88 ÷ 11 = 8

= 2 × 2 × 2

= 2³

∴ The smallest number by which the number 88 must be divided to obtain a perfect cube is 11.

Question 9.

Choose the correct answer for the following:

The volume of a cube is 64 cm³. The side of the cube is
A. 4 cm

B. 8 cm

C. 16 cm

D. 6 cm

Answer:

We know that the Volume of a cube = a³ where a is the side of the cube.

But given Volume of cube = 64 cm³

⇒ a³ = 64

Prime Factorization of 64:

⇒ 64 = 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³

= 4³

∴ a³ = 4³

Powers are equal, so bases must be equated.

∴ a = 4 cm (Side of the cube)

Question 10.

Choose the correct answer for the following:

Which of the following is false?
A. Cube of any odd number is odd.

B. A perfect cube does not end with two zeros.

C. The cube of a single digit number may be a single digit number.

D. There is no perfect cube which ends with 8.

Answer:

A. We know that cubes of odd numbers are all odd numbers.

∴ This is true.

B. For example:

Cube of 10 = 10³

= 10 × 10 × 10

= 1000

Which ends in 3 zeros.

∴ It is true that a perfect cube does not end with two zeros.

C. For example:

Cube of 2 = 2³

= 2 × 2 × 2

= 8 (single digit)

While cube of 3 = 3³

= 3 × 3 × 3

= 27 (double digit)

∴ It can be concluded that the cube of a single digit number may be a single digit number.

D. For example:

2³ = 8; 12³ = 1728; 22³ = 10648

Here, we can see that there are cubes which end with the digit 8.

∴ It is false that there is no perfect cube which ends with 8.

Question 11.

Check whether the following are perfect cubes?

400

Answer:

400

Prime Factorization:

⇒ 400 = 2 × 2 × 2 × 2 × 5 × 5

= 2³ × 2 × 5²

There is only one 1 and two 5’s.

∴ 400 is not a perfect cube.

Question 12.

Check whether the following are perfect cubes?

216

Answer:

216

Prime Factorization:

⇒ 216 = 2 × 2 × 2 × 3 × 3 × 3

= 2³ × 3³

= (2 × 3)³

= 6³

∴ 216 is a perfect cube.

Question 13.

Check whether the following are perfect cubes?

729

Answer:

729

Prime Factorization:

⇒ 729 = 3 × 3 × 3 × 3 × 3 × 3

= 3³ × 3³

= (3 × 3)³

= 9³

∴ 729 is a perfect cube.

Question 14.

Check whether the following are perfect cubes?

250

Answer:

250

Prime Factorization:

⇒ 250 = 2 × 5 × 5 × 5

There is only one 2 in the factorization.

∴ 250 is not a perfect cube.

Question 15.

Check whether the following are perfect cubes?

1000

Answer:

1000

Prime Factorization:

⇒ 1000 = (2 × 2 × 2) × (5 × 5 × 5)

= 2³ × 5³

= (2 × 5)³

= 10³

∴ 1000 is a perfect cube.

Question 16.

Check whether the following are perfect cubes?

900

Answer:

900

Prime Factorization:

⇒ 900 = 3 × 3 × 2 × 2 × 5 × 5

There are only two 3’s, 2’s and 5’s.

∴ 900 is not a perfect cube.

∴ ii, iii and v are perfect cubes.

Question 17.

Which of the following numbers are not perfect cubes?

128

Answer:

128

Prime Factorization:

⇒ 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³ × 2

There is only one 2.

∴ 128 is not a perfect cube.

Question 18.

Which of the following numbers are not perfect cubes?

100

Answer:

100

Prime Factorization:

⇒ 100 = 2 × 2 × 5 × 5

There are only two 2’s and two 5’s.

Hence, 100 is not a perfect cube.

Question 19.

Which of the following numbers are not perfect cubes?

64

Answer:

64

Prime Factorization:

⇒ 64 = 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³

= 4³

∴ 64 is a perfect cube.

Question 20.

Which of the following numbers are not perfect cubes?

125

Answer:

125

Prime factorization:

⇒ 125 = 5 × 5 × 5 = 5³

∴ 125 is a perfect cube.

Question 21.

Which of the following numbers are not perfect cubes?

72

Answer:

72

Prime Factorization:

⇒ 72 = 2 × 2 × 2 × 3 × 3

= 2³ × 3²

There are only two 3’s.

∴ 72 is not a perfect cube.

Question 22.

Which of the following numbers are not perfect cubes?

625

Answer:

625

Prime Factorization:

⇒ 625 = 5 × 5 × 5 × 5

= 5³ × 5

There is only one 5.

∴ 625 is not a perfect cube.

∴ i, ii, v, vi are not perfect cubes.

Question 23.

Find the smallest number by which each of the following number must be divided to obtain a perfect cube.

81

Answer:

81

Prime Factorization:

⇒ 81 = 3 × 3 × 3 × 3

= 3³ × 3

There is only one 3.

∴ 81 is not a perfect cube.

Since in factorization, 3 appear only one time, we should divide the number 81 by 3.

⇒ 81 ÷ 3 = 27

= 3 × 3 × 3

= 3³ which is a perfect cube

∴ The smallest number by which the number 81 must be divided to obtain a perfect cube is 3.

Question 24.

Find the smallest number by which each of the following number must be divided to obtain a perfect cube.

128

Answer:

128

Prime Factorization:

⇒ 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³ × 2

There is only one 2.

∴ 128 is not a perfect cube.

Since in factorization, 2 appear only one time, we should divide the number 128 by 2.

⇒ 128 ÷ 2 = 64

= 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³

= 4³ which is a perfect cube

∴ The smallest number by which the number 128 must be divided to obtain a perfect cube is 2.

Question 25.

Find the smallest number by which each of the following number must be divided to obtain a perfect cube.

135

Answer:

135

Prime Factorization:

⇒ 135 = 3 × 3 × 3 × 5

= 3³ × 5

There is only one 5.

∴ 135 is not a perfect cube.

Since in factorization, 5 appear only one time, we should divide the number 135 by 5.

⇒ 135 ÷ 5 = 27

= 3 × 3 × 3

= 3³ which is a perfect cube

∴ The smallest number by which the number 135 must be divided to obtain a perfect cube is 5.

Question 26.

Find the smallest number by which each of the following number must be divided to obtain a perfect cube.

192

Answer:

192

Prime Factorization:

⇒ 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3

= 2³ × 2³ × 3

There is only one 3.

∴ 192 is not a perfect cube.

Since in factorization, 3 appear only one time, we should divide the number 192 by 3.

⇒ 192 ÷ 3 = 64

= 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³

= 4³ which is a perfect cube

∴ The smallest number by which the number 192 must be divided to obtain a perfect cube is 3.

Question 27.

Find the smallest number by which each of the following number must be divided to obtain a perfect cube.

704

Answer:

704

Prime Factorization:

⇒ 704 = 2 × 2 × 2 × 2 × 2 × 2 × 11

= 2³ × 2³ × 11

There is only one 11.

∴ 704 is not a perfect cube.

Since in factorization, 11 appear only one time, we should divide the number 704 by 11.

⇒ 704 ÷ 11 = 64

= 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³

= 4³ which is a perfect cube

∴ The smallest number by which the number 704 must be divided to obtain a perfect cube is 11.

Question 28.

Find the smallest number by which each of the following number must be divided to obtain a perfect cube.

625

Answer:

625

Prime Factorization:

⇒ 625 = 5 × 5 × 5 × 5

= 5³ × 5

There is only one 5.

∴ 625 is not a perfect cube.

Since in factorization, 5 appear only one time, we should divide the number 625 by 5.

⇒ 625 ÷ 5 = 125

= 5 × 5 × 5

= 5³

∴ The smallest number by which the number 625 must be divided to obtain a perfect cube is 5.

Question 29.

Find the smallest number by which each of the following number must be multiplied to obtain a perfect cube.

243

Answer:

243

Prime Factorization:

⇒ 243 = 3 × 3 × 3 × 3 × 3

= 3³ × 3²

There are only two 3’s.

∴ 243 is not a perfect cube.

To make it a perfect cube, we multiply it with 3.

⇒ 243 × 3 = 729

= 3 × 3 × 3 × 3 × 3 × 3

= 3³ ×3³ which is a perfect cube

∴ The smallest number by which the number 243 must be multiplied to obtain a perfect cube is 3.

Question 30.

Find the smallest number by which each of the following number must be multiplied to obtain a perfect cube.

256

Answer:

256

Prime Factorization:

⇒ 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³ × 2²

There are only two 2’s.

∴ 256 is not a perfect cube.

To make it a perfect cube, we multiply with 2.

⇒ 256 × 2 = 512

= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³ × 2³

= 8³ which is a perfect cube

∴ The smallest number by which the number 256 must be multiplied to obtain a perfect cube is 2.

Question 31.

Find the smallest number by which each of the following number must be multiplied to obtain a perfect cube.

72

Answer:

72

Prime Factorization:

⇒ 72 = 2 × 2 × 2 × 3 × 3

= 2³ × 3²

There are only two 3’s.

∴ 72 is not a perfect cube.

To make it a perfect cube, we have to multiply with 3.

⇒ 72 × 3 = 2 × 2 × 2 × 3 × 3 × 3

⇒ 216 = 2³ × 3³

= 6³ which is a perfect cube.

∴ The smallest number by which 72 must be multiplied to obtain a perfect cube is 3.

Question 32.

Find the smallest number by which each of the following number must be multiplied to obtain a perfect cube.

675

Answer:

675

Prime Factorization:

⇒ 675 = 5 × 5 × 3 × 3 × 3

= 3³ × 5²

There are only two 5’s.

∴ 675 is not a perfect cube.

To make it a perfect cube, we multiply with 5.

⇒ 675 × 5 = 3375

= 5 × 5 × 5 × 3 × 3 × 3

= 3³ × 5³

= 15³ which is a perfect cube

∴ The smallest number by which the number 675 must be multiplied to obtain a perfect cube is 5.

Question 33.

Find the smallest number by which each of the following number must be multiplied to obtain a perfect cube.

100

Answer:

100

Prime Factorization:

⇒ 100 = 2 × 2 × 5 × 5

There are only two 2’s and two 5’s.

Hence, 100 is not a perfect cube.

To make it a perfect cube, we have to multiply it with 2 × 5 = 10.

⇒ 100 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5

⇒ 1000 = 2³ × 5³

= 10³ which is a perfect cube.

∴ The smallest number by which 100 must be multiplied to obtain a perfect cube is 10.

Question 34.

Find the cube root of each of the following numbers by prime Factorization method:

729

Answer:

729

Prime Factorization:

⇒  = (729)^1/3

= ((3 × 3 × 3) × (3 × 3 × 3))^1/3

= (3³ × 3³)^1/3

= (9³)^1/3

We know that by laws of exponents, (a^m)ⁿ = a^mn.

∴  = 9

Question 35.

Find the cube root of each of the following numbers by prime Factorization method:

343

Answer:

343

Prime Factorization:

⇒  = (343)^1/3

= (7 × 7 × 7)^1/3

= (7³)^1/3

We know that by laws of exponents, (a^m)ⁿ = a^mn.

∴  = 7

Question 36.

Find the cube root of each of the following numbers by prime Factorization method:

512

Answer:

512

Prime Factorization:

⇒  = (512)^1/3

= ((2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2))^1/3

= (2³ × 2³ × 2³)^1/3

= (8³)^1/3

We know that by laws of exponents, (a^m)ⁿ = a^mn.

∴  = 8

Question 37.

Find the cube root of each of the following numbers by prime Factorization method:

0.064

Answer:

0.064

⇒  = 

= 

= 

= 

= 

= 

= 0.4

∴  = 0.4

Question 38.

Find the cube root of each of the following numbers by prime Factorization method:

0.216

Answer:

0.216

⇒  = 

= 

= 

= 

= 

= 

= 0.6

∴  = 0.6

Question 39.

Find the cube root of each of the following numbers by prime Factorization method:

Answer:

 can be written as 

⇒  = 

= 

= 

= 

= 

= 1.75

∴  = 1.75

Question 40.

Find the cube root of each of the following numbers by prime Factorization method:

– 1.331

Answer:

-1.331

⇒  = 

= 

= 

= 

= 

We know that cube root of a negative number is negative.

= 

= -1.1

∴  = -1.1

Question 41.

Find the cube root of each of the following numbers by prime Factorization method:

– 27000

Answer:

-27000

-27000 can be written as -27 × 1000.

Prime Factorization of 27:

⇒ 27 = 3 × 3 × 3

= 3³

Prime Factorization of 1000:

⇒ 1000 = (2 × 2 × 2) × (5 × 5 × 5)

= (2³ × 5³)

= 10³

⇒  = (-27 × 1000)^1/3

= (-3³ × 10³)^1/3

= (-30³)^1/3

We know that by laws of exponents, (a^m)ⁿ = a^mn.

We know that cube root of a negative number is negative.

∴  = -30

Question 42.

The volume of a cubical box is 19.683 cu. cm. Find the length of each side of the box.

Answer:

We know that the Volume of a cube = a³ where a is the side of the cube.

But given Volume of cube = 19.683 cm³

⇒ a³ = 19.683

⇒ a = 

⇒  = 

= 

= 

= 

= 

= 

= 0.9

∴  = 0.9 = a

∴ The length of each side of the box = 0.9 cm.

---

Exercise 1.8

Question 1.

Express the following correct to two decimal places:

12.568

Answer:

12.568

It is 12.57 correct to two decimal places.

Since the last digit 8 > 5, we add 1 to 6 and make it 7.

∴ 12.568 ≈ 12.57 (correct to two decimal places)

Question 2.

Express the following correct to two decimal places:

25.416 kg

Answer:

25.416 kg

It is 25.42 kg correct to two decimal places.

Since the last digit 6 > 5, we add 1 to 1 and make it 2.

∴ 25.416 ≈ 25.42 kg (correct to two decimal places)

Question 3.

Express the following correct to two decimal places:

39.927 m

Answer:

39.927 m

It is 39.93 m correct to two decimal places.

Since the last digit 7 > 5, we add 1 to 2 and make it 3.

∴ 39.927 ≈ 39.93 m (correct to two decimal places)

Question 4.

Express the following correct to two decimal places:

56.596 m

Answer:

56.596 m

It is 56.60 m correct to two decimal places.

Since the last digit 6 > 5, we add 1 to 59 and make it 60.

∴ 56.596 ≈ 56.60 m (correct to two decimal places)

Question 5.

Express the following correct to two decimal places:

41.056 m

Answer:

41.056 m

It is 41.06 m correct to two decimal places.

Since the last digit 6 > 5, we add 1 to 5 and make it 6.

∴ 41.056 ≈ 41.06 m (correct to two decimal places)

Question 6.

Express the following correct to two decimal places:

729.943 km

Answer:

729.943 km

It is 729.94 km correct to two decimal places.

Since the last digit 3 < 5, so we leave 4 as it is.

∴ 729.943 ≈ 729.94 km (correct to two decimal places)

Question 7.

Express the following correct to three decimal places:

0.0518 m

Answer:

0.0518 m

It is 0.052 m correct to three decimal places.

Since the last digit 8 > 5, we add 1 to 1 and make it 2.

∴ 0.0518 ≈ 0.052 m (correct to three decimal places)

Question 8.

Express the following correct to three decimal places:

3.5327 km

Answer:

3.5327 km

It is 3.533 km correct to three decimal places.

Since the last digit 7 > 5, we add 1 to 2 and make it 3.

∴ 3.5327 ≈ 3.533 km (correct to three decimal places)

Question 9.

Express the following correct to three decimal places:

58.2936l

Answer:

58.2936 l

It is 58.294 l correct to three decimal places.

Since the last digit 6 > 5, we add 1 to 3 and make it 4.

∴ 58.2936 ≈ 58.294 l (correct to three decimal places)

Question 10.

Express the following correct to three decimal places:

0.1327 gm

Answer:

0.1327 gm

It is 0.133 gm correct to three decimal places.

Since the last digit 7 > 5, we add 1 to 2 and make it 3.

∴ 0.1327 ≈ 0.133 gm (correct to three decimal places)

Question 11.

Express the following correct to three decimal places:

365.3006

Answer:

365.3006

It is 365.301 correct to three decimal places.

Since the last digit 6 > 5, we add 1 to 0 and make it 1.

∴ 365.3006 ≈ 365.301 (correct to three decimal places)

Question 12.

Express the following correct to three decimal places:

100.1234

Answer:

100.1234

It is 100.123 correct to three decimal places.

Since the last digit 4 < 5, so we leave 3 as it is.

∴ 100.1234 ≈ 100.123 (correct to three decimal places)

Question 13.

Write the approximate value of the following numbers to the accuracy stated:

247 to the nearest ten.

Answer:

247 to the nearest ten.

Consider multiples of 10 before and after 247 (i.e. 240 and 250).

We find that 247 is nearer to 250 than to 240.

∴ The approximate value of 247 is 250.

Question 14.

Write the approximate value of the following numbers to the accuracy stated:

152 to the nearest ten.

Answer:

152 to the nearest ten.

Consider multiples of 10 before and after 152 (i.e. 150 and 160).

We find that 152 is nearer to 150 than to 160.

∴ The approximate value of 152 is 150.

Question 15.

Write the approximate value of the following numbers to the accuracy stated:

6848 to the nearest hundred.

Answer:

6848 to the nearest hundred.

Consider multiples of 100 before and after 6848 (i.e. 6800 and 6900).

We find that 6848 is nearer to 6800 than to 6900.

∴ The approximate value of 6848 is 6800.

Question 16.

Write the approximate value of the following numbers to the accuracy stated:

14276 to the nearest ten thousand.

Answer:

14276 to the nearest ten thousand.

Consider multiples of 10, 000 before and after 14276 (i.e. 10, 000 and 20, 000).

We find that 14276 is nearer to 10, 000 than to 20, 000.

∴ The approximate value of 14276 is 10, 000.

Question 17.

Write the approximate value of the following numbers to the accuracy stated:

3576274 to the nearest Lakhs.

Answer:

3576274 to the nearest Lakhs.

Consider multiples of 1 lakh (1, 00, 000) before and after 3576274 (i.e. 35 lakhs and 36 lakhs).

We find that 3576274 is nearer to 36, 00, 000 than to 35, 00, 000.

∴ The approximate value of 3576274 is 36, 00, 000 i.e. 36 lakhs.

Question 18.

Write the approximate value of the following numbers to the accuracy stated:

104, 3567809 to the nearest crore

Answer:

104, 3567809 to the nearest crore

Consider multiples of 1 crore (1, 00, 00, 000) before and after 104, 3567809 (i.e. 104 crores and 105 crores).

We find that 104, 3567809 is nearer to 104 crores than to 105 crores.

∴ The approximate value of 104, 3567809 is 104 crores.

Question 19.

Round off the following numbers to the nearest integer:

22.266

Answer:

i. 22.266

Here, the hundredth place 2 < 5, so the number is left as it is.

∴ 22.266 ≈ 22

Question 20.

Round off the following numbers to the nearest integer:

777.43

Answer:

777.43

Here, the tenth place 4 < 5, so the number is left as it is.

∴ 777.43 ≈ 777

Question 21.

Round off the following numbers to the nearest integer:

402.06

Answer:

402.06

Here, the tenth place 0 < 5, so the number is left as it is.

∴ 402.06 ≈ 402

Question 22.

Round off the following numbers to the nearest integer:

305.85

Answer:

305.85

Here, the tenth place 8 > 5, so the integer value is increased by 1.

∴ 305.85 ≈ 306

Question 23.

Round off the following numbers to the nearest integer:

299.77

Answer:

299.77

Here, the tenth place 7 > 5, so the integer value is increased by 1.

∴ 299.77 ≈ 300

Question 24.

Round off the following numbers to the nearest integer:

9999.9567

Answer:

9999.9567

Here, the thousandth place 9 > 5, so the integer value is increased by 1.

∴ 9999.9567 ≈ 10000

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Exercise 1.9

Question 1.

Complete the following patterns:

40, 35, 30, ___, ___, ___.

Answer:

40, 35, 30, ___, ___, ___.

Here, each term is 5 less than the previous term.

⇒ The next three terms are: 30 – 5 = 25

25 – 5 = 20

20 – 5 = 15

∴ The pattern is 40, 35, 30, 25, 20, 15.

Question 2.

Complete the following patterns:

0, 2, 4, ___, ___, ___.

Answer:

0, 2, 4, ___, ___, ___.

Here, the terms are even numbers.

⇒ The next three terms are: 6, 8, 10.

∴ The pattern is 0, 2, 4, 6, 8, 10.

Question 3.

Complete the following patterns:

84, 77, 70, ___, ___, ___.

Answer:

84, 77, 70, ___, ___, ___.

Here, each term is 7 less than the previous term (or) multiples of 7 in decreasing order starting from 12.

⇒ The next three terms are: 70 – 7 = 63

63 – 7 = 56

56 – 7 = 49

∴ The pattern is 84, 77, 70, 63, 56, 49.

Question 4.

Complete the following patterns:

4.4, 5.5, 6.6, ___, ___, ___.

Answer:

4.4, 5.5, 6.6, ___, ___, ___.

Here, each term is 1.1 more than the previous term.

⇒ The next three terms are: 6.6 + 1.1 = 7.7

7.7 + 1.1 = 8.8

8.8 + 1.1 = 9.9

∴ The pattern is 4.4, 5.5, 6.6, 7.7, 8.8, 9.9.

Question 5.

Complete the following patterns:

1, 3, 6, 10, ___, ___, ___.

Answer:

1, 3, 6, 10, ___, ___, ___.

Here, each term is (n + 1) more than the previous term starting from n = 2.

⇒ The next three terms are: 10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

∴ The pattern is 1, 3, 6, 10, 15, 21, 28.

Question 6.

Complete the following patterns:

1, 1, 2, 3, 5, 8, 13, 21, ___, ___, ___

(This sequence is called FIBONACCI SEQUENCE)

Answer:

1, 1, 2, 3, 5, 8, 13, 21, ___, ___, ___

(This sequence is called FIBONACCI SEQUENCE)

Here, the series starts with 1 and the sum of every subsequent term is the sum of previous two.

⇒ The next three terms are: 21 + 13 = 34

34 + 21 = 55

55 + 34 = 89

∴ The pattern is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.

Question 7.

Complete the following patterns:

1, 8, 27, 64, ___, ___, ___.

Answer:

1, 8, 27, 64, ___, ___, ___.

Here, the terms are cubes of natural numbers.

⇒ The next three terms are: 5³ = 125

6³ = 216

7³ = 343

∴ The pattern is 1, 8, 27, 64, 125, 216, 343.

Question 8.

A water tank has steps inside it. A monkey is sitting on the top most step. (ie, the first step) The water level is at the ninth step.

A. He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level?



B. After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step?

Answer:

Let the steps moved down be represented by positive integers and the steps moved up be the negative integers.

A. First, the monkey is at = 1^st step

∴ The monkey will be at water level i.e. 9^th step after 11 jumps.

B. Now, the monkey is at = 9^th step

∴ The monkey will reach back at the top step after 5 jumps.

Question 9.

A vendor arranged his apples as in the following pattern:

A. If there are ten rows of apples, can you find the total number of apples without actually counting?

B. If there are twenty rows, how many apples will be there in all?



Can you recognize a pattern for the total number of apples? Fill this chart and try!

Answer:

Given in the pattern of apples arranged,

1^st row = 1 apple, 2^nd row = 2 apples, 3^rd row = 3 apples and 4^th row = 4 apples and so on.

So, this can be expressed as 1 + 2 + 3 + 4 + …

We know that sum of numbers from 1 to n is .

A. We have to find the total number of apples in 10 rows.

Here n = 10.

∴ Number of apples in 10 rows =  =  = 55 apples

B. We have to find the total number of apples in 20 rows.

Here n = 20.

∴ Number of apples in 20 rows =  =  = 210 apples

Here, the series starts with 1 and the sum of every subsequent term is the sum of previous two. (Fibonacci series)