Rs Aggarwal 2018 for Class 8 Math Chapter 4 - Cubes And Cube Roots

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Page No 64:

Question 1:

Evaluate:
(i) (8)³
(ii) (15)³
(iii) (21)³
(iv) (60)³

Answer:

(i) (8)³ = $(8 \times 8 \times 8)$ = 512.
Thus, the cube of 8 is 512.
(ii) (15)³= $(15 \times 15 \times 15)$ = 3375.
Thus, the cube of 15 is 3375.
(iii) (21)³= $(21 \times 21 \times 21)$ = 9261.
Thus, the cube of 21 is 9261.
(iv) (60)³= $(60 \times 60 \times 60)$ = 216000.
Thus, the cube of 60 is 216000.

Page No 64:

Question 2:

Evaluate:
(i) (1.2)3
(ii) (3.5)³
(iii) (0.8)³
(iv) (0.05)³

Answer:

(i) (1.2)³ = $(1.2 \times 1.2 \times 1.2)$ = 1.728
Thus, the cube of 1.2 is 1.728.
(ii) (3.5)³= $(3.5 \times 3.5 \times 3.5)$ = 42.875
Thus, the cube of 3.5 is 42.875.
(iii) (0.8)³= $(0.8 \times 0.8 \times 0.8)$ = 0.512
Thus, the cube of 0.8 is 0.512.
(iv) (0.05)³= $(0.05 \times 0.05 \times 0.05)$ = 0.000125
Thus, the cube of 0.05 is 0.000125.

Page No 65:

Question 3:

Evaluate:
(i) ${(\frac{4}{7})}^{3}$
(ii) ${(\frac{10}{11})}^{3}$
(iii) ${(\frac{1}{15})}^{3}$
(iv) ${(1 \frac{3}{10})}^{3}$

Answer:

(i) ${(\frac{4}{7})}^{3}$ = $(\frac{4}{7} \times \frac{4}{7} \times \frac{4}{7})$ = $(\frac{64}{343})$
Thus, the cube of $(\frac{4}{7})$ is $(\frac{64}{343})$ .

(ii) ${(\frac{10}{11})}^{3}$ = $(\frac{10}{11} \times \frac{10}{11} \times \frac{10}{11})$ = $(\frac{1000}{1331})$
Thus, the cube of $(\frac{10}{11})$ is $(\frac{1000}{1331})$ .(47)3
(iii) ${(\frac{1}{15})}^{3}$ = $(\frac{1}{15} \times \frac{1}{15} \times \frac{1}{15})$ = $(\frac{1}{3375})$
Thus, the cube of $(\frac{1}{15})$ is $(\frac{1}{3375})$
(iv) ${(1 \frac{3}{10})}^{3}$ = ${(\frac{13}{10})}^{3}$ = $(\frac{13}{10} \times \frac{13}{10} \times \frac{13}{10})$ = $(\frac{2197}{1000})$
Thus, the cube of $(1 \frac{3}{10})$ is $(\frac{2197}{1000})$ . (47)3

Page No 65:

Question 4:

Which of the following numbers are perfect cubes? In case of perfect cube, find the number whose cube is the given number.
(i) 125
(ii) 243
(iii) 343
(iv) 256
(v) 8000
(vi) 9261
(vii) 5324
(viii) 3375

Answer:

(i) 125
Resolving 125 into prime factors:
125 = 5 $\times$ 5 $\times$ 5
Here, one triplet is formed, which is $5^{3}$ . Hence, 125 can be expressed as the product of the triplets of 5.
Therefore, 125 is a perfect cube.

(ii) 243 is not a perfect cube.

(iii) 343
Resolving 125 into prime factors:
343 = 7 $\times$ 7 $\times$ 7
Here, one triplet is formed, which is $7^{3}$ . Hence, 343 can be expressed as the product of the triplets of 7.
Therefore, 343 is a perfect cube.

(iv) 256 is not a perfect cube.

(v) 8000
Resolving 8000 into prime factors:
8000 = 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 5 $\times$ 5 $\times$ 5
Here, three triplets are formed, which are 2³, 2³ and 5³. Hence, 8000 can be expressed as the product of the triplets of 2, 2 and 5, i.e. $2^{3}$ $\times$ $2^{3}$ $\times$ $5^{3}$ = $20^{3}$ .
Therefore, 8000 is a perfect cube.

(vi) 9261
Resolving 9261 into prime factors:
9261 = 3 $\times$ 3 $\times$ 3 $\times$ 7 $\times$ 7 $\times$ 7
Here, two triplets are formed, which are $3^{3}$ and $7^{3}$ . Hence, 9261 can be expressed as the product of the triplets of 3 and 7, i.e. $3^{3}$ $\times$ $7^{3}$ = $21^{3}$ .
Therefore, 9261 is a perfect cube.

(vii) 5324 is not a perfect cube.

(viii) 3375 .
Resolving 3375 into prime factors:
3375 = 3 $\times$ 3 $\times$ 3 $\times$ 5 $\times$ 5 $\times$ 5.
Here, two triplets are formed, which are $3^{3}$ and $5^{3}$ . Hence, 3375 can be expressed as the product of the triplets of 3 and 5, i.e. $3^{3}$ $\times$ $5^{3}$ = $15^{3}$ .
Therefore, 3375 is a perfect cube.

Page No 65:

Question 5:

Which of the following are the cubes of even numbers?
(i) 216
(ii) 729
(iii) 512
(iv) 3375
(v) 1000

Answer:

The cubes of even numbers are always even. Therefore, 216, 512 and 1000 are the cubes of even numbers.

216 = 2 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 3 $\times$ 3 = $2^{3} \times 3^{3} = 6^{3}$
512 = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{3} \times 2^{3} \times 2^{3} = 8^{3}$
1000 = $2 \times 2 \times 2 \times 5 \times 5 \times 5 = 2^{3} \times 5^{3} = 10^{3}$

Page No 65:

Question 6:

Which of the following are the cubes of odd numbers?
(i) 125
(ii) 343
(iii) 1728
(iv) 4096
(v) 9261

Answer:

The cube of an odd number is an odd number. Therefore, 125, 343 and 9261 are the cubes of odd numbers.

125 = 5 $\times$ 5 $\times$ 5 = $5^{3}$

343 = 7 $\times$ 7 $\times$ 7 = $7^{3}$

9261 = 3 $\times$ 3 $\times$ 3 $\times$ 7 $\times$ 7 $\times$ 7 = $3^{3}$ $\times$ $7^{3}$ = $21^{3}$

Page No 65:

Question 7:

Find the smallest number by which 1323 must be multiplied so that the product is a perfect cube.

Answer:

1323

1323 = $3 \times 3 \times 3 \times 7 \times 7$ .
To make it a perfect cube, it has to be multiplied by 7.

Page No 65:

Question 8:

Find the smallest number by which 2560 must be multiplied so that the product is a perfect cube.

Answer:

2560

2560 can be expressed as the product of prime factors in the following manner:

2560 = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5$

To make this a perfect square, we have to multiply it by 5 $\times$ 5.
Therefore, 2560 should be multiplied by 25 so that the product is a perfect cube.

Page No 65:

Question 9:

What is the smallest number by which 1600 must be divided so that the quotient is a perfect cube?

Answer:

1600

1600 can be expressed as the product of prime factors in the following manner:

1600 = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5$

Therefore, to make the quotient a perfect cube, we have to divide 1600 by:
$5 \times 5 = 25$

Page No 65:

Question 10:

Find the smallest number by which 8788 must be divided so that the quotient is a perfect cube.

Answer:

8788
8788 can be expressed as the product of prime factors as $2 \times 2 \times 13 \times 13 \times 13$ .
Therefore, 8788 should be divided by 4, i.e. ( $2 \times 2$ ), so that the quotient is a perfect cube.

Page No 66:

Question 1:

Find the value of using the short-cut method:
(25)³

Answer:

${(25)}^{3}$
Here, a = 2 and b = 5

Using the formula $a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ :

4 $\times$ 2	4 $\times$ 15	25 $\times$ 6	25 $\times$ 5
8 $+$ 7	60 $+$ 16	150 $+$ 12	125
15	76	162

∴

{(25)}^{3}

= 15625

Page No 66:

Question 2:

Find the value of using the short-cut method:
(47)³

Answer:

${(47)}^{3}$
Here, a = 4 and b = 7

Using the formula $a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ :

16 $\times$ 4	16 $\times$ 21	49 $\times$ 12	49 $\times$ 7
64 $+$ 39	336 $+$ 62	588 $+$ 34	343
103	398	622

∴

{(47)}^{3}

= 103823

Page No 66:

Question 3:

Find the value of using the short-cut method:
(68)³

Answer:

${(68)}^{3}$
Here, a = 6 and b = 8

Using the formula $a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ :

36 $\times$ 6	36 $\times$ 24	64 $\times$ 18	64 $\times$ 8
216 $+$ 98	864 $+$ 120	1152 $+$ 51	512
314	984	1203

∴

{(68)}^{3}

= 314432

Page No 66:

Question 4:

Find the value of using the short-cut method:
(84)³

Answer:

${(84)}^{3}$

Here, a = 8 and b = 4

Using the formula $a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ :

64 $\times$ 8	64 $\times$ 12	16 $\times$ 24	16 $\times$ 4
512 $+$ 80	768 $+$ 39	384 $+$ 6	64
592	807	390

∴

{(84)}^{3}

= 592704

Page No 67:

Question 1:

Evaluate:
$\sqrt[3]{64}$

Answer:

$\sqrt[3]{64}$
By prime factorisation:
64 = $2 \times 2 \times 2 \times 2 \times 2 \times 2$
= $(2 \times 2 \times 2) \times (2 \times 2 \times 2)$

∴ $\sqrt[3]{64} = \sqrt[3]{(2)^{3} \times (2)^{3}} = (2 \times 2) = 4$

Page No 67:

Question 2:

Evaluate:
$\sqrt[3]{343}$

Answer:

$\sqrt[3]{343}$
By prime factorisation:
343 = $7 \times 7 \times 7$
= ( $7 \times 7 \times 7$ )

∴ $\sqrt[3]{343} = \sqrt[3]{7^{3}} = 7$

Page No 67:

Question 3:

Evaluate:
$\sqrt[3]{729}$

Answer:

$\sqrt[3]{729}$
By prime factorisation:

729 = $3 \times 3 \times 3 \times 3 \times 3 \times 3$
= $(3 \times 3 \times 3) \times (3 \times 3 \times 3)$

∴ $\sqrt[3]{729}$ = $(3 \times 3) = 9$

Page No 67:

Question 4:

Evaluate:
$\sqrt[3]{1728}$

Answer:

$\sqrt[3]{1728}$
By prime factorisation:

1728 = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$
= $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3) = 2^{3} \times 2^{3} \times 3^{3}$

∴ $\sqrt[3]{1728}$ = $(2 \times 2 \times 3) = 12$

Page No 67:

Question 5:

Evaluate:
$\sqrt[3]{9261}$

Answer:

$\sqrt[3]{9261}$
By prime factorisation:

9261 = $3 \times 3 \times 3 \times 7 \times 7 \times 7$
= $(3 \times 3 \times 3) \times (7 \times 7 \times 7) = 3^{3} \times 7^{3}$

∴ $\sqrt[3]{9261}$ = $(3 \times 7) = 21$

Page No 67:

Question 6:

Evaluate:
$\sqrt[3]{4096}$

Answer:

$\sqrt[3]{4096}$
By prime factorisation:

4096 = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
= $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) = 2^{3} \times 2^{3} \times 2^{3} \times 2^{3}$

∴ $\sqrt[3]{4096}$ = $(2 \times 2 \times 2 \times 2) = 16$

Page No 67:

Question 7:

Evaluate:
$\sqrt[3]{8000}$

Answer:

$\sqrt[3]{8000}$
By prime factorisation:

8000 = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5$
= $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (5 \times 5 \times 5)$

∴ $\sqrt[3]{8000}$ = $(2 \times 2 \times 5) = 20$

Page No 67:

Question 8:

Evaluate:
$\sqrt[3]{3375}$

Answer:

$\sqrt[3]{3375}$
By prime factorisation:

3375 = $3 \times 3 \times 3 \times 5 \times 5 \times 5$
= $(3 \times 3 \times 3) \times (5 \times 5 \times 5)$

∴ $\sqrt[3]{3375}$ = $(3 \times 5) = 15$

Page No 68:

Question 9:

Evaluate:
$\sqrt[3]{- 216}$

Answer:

$\sqrt[3]{- 216}$
By prime factorisation:

216 = $2 \times 2 \times 2 \times 3 \times 3 \times 3$
= $(2 \times 2 \times 2) \times (3 \times 3 \times 3)$
$\sqrt[3]{- 216}$ = $- (2 \times 3) = - 6$

∴ $\sqrt[3]{- 216}$ = $- (\sqrt[3]{216}) = - 6$

Page No 68:

Question 10:

Evaluate:
$\sqrt[3]{- 512}$

Answer:

$\sqrt[3]{- 512}$
By prime factorisation:

$\sqrt[3]{512}$ = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
= $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)$
$\sqrt[3]{- 512}$ = $- \sqrt[3]{(2 \times 2 \times 2)} = - 8$

∴ $\sqrt[3]{- 512}$ = $- (\sqrt[3]{512}) = - 8$

Page No 68:

Question 11:

Evaluate:
$\sqrt[3]{- 1331}$

Answer:

$\sqrt[3]{- 1331}$
By prime factorisation:
$\sqrt[3]{1331}$ = $\sqrt[3]{11 \times 11 \times 11}$

$\sqrt[3]{- 1331}$ = $- {(11 \times 11 \times 11)}^{\frac{1}{3}} = - 11$
∴ $\sqrt[3]{- 1331} = - (\sqrt[3]{1331}) = - 11$

Page No 68:

Question 12:

Evaluate:
$\sqrt[3]{\frac{27}{64}}$

Answer:

$\sqrt[3]{\frac{27}{64}}$
By prime factorisation:

$\sqrt[3]{\frac{27}{64}}$ = $\frac{\sqrt[3]{27}}{\sqrt[3]{64}}$ = $\frac{\sqrt[3]{(3 \times 3 \times 3)}}{\sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2)}}$ = $\frac{\sqrt[3]{(3 \times 3 \times 3)}}{\sqrt[3]{(4 \times 4 \times 4)}} = \frac{3}{4}$
∴ $\sqrt[3]{\frac{27}{64}}$ = $\frac{3}{4}$

Page No 68:

Question 13:

Evaluate:
$\sqrt[3]{\frac{125}{216}}$

Answer:

$\sqrt[3]{\frac{125}{216}}$
By prime factorisation:

$\sqrt[3]{\frac{125}{216}}$ = $\frac{\sqrt[3]{5 \times 5 \times 5}}{\sqrt[3]{(2 \times 2 \times 2) \times (3 \times 3 \times 3)}}$ = $\frac{\sqrt[3]{5 \times 5 \times 5}}{\sqrt[3]{(6 \times 6 \times 6)}} = \frac{5}{6}$

∴ $\sqrt[3]{\frac{125}{216}}$ = $\frac{5}{6}$

Page No 68:

Question 14:

Evaluate:
$\sqrt[3]{\frac{- 27}{125}}$

Answer:

$\sqrt[3]{\frac{- 27}{125}}$

By factorisation:
$\sqrt[3]{\frac{27}{125}}$ = $\sqrt[3]{\frac{3 \times 3 \times 3}{5 \times 5 \times 5}}$

∴ $\sqrt[3]{\frac{- 27}{125}}$ = $\frac{- 3}{5}$

Page No 68:

Question 15:

Evaluate:
$\sqrt[3]{\frac{- 64}{343}}$

Answer:

$\sqrt[3]{\frac{- 64}{343}}$
On factorisation:

$\sqrt[3]{\frac{64}{343}}$ = $\sqrt[3]{\frac{2 \times 2 \times 2 \times 2 \times 2 \times 2}{7 \times 7 \times 7}}$
∴ $\sqrt[3]{\frac{- 64}{343}}$ = $\frac{- 4}{7}$

Page No 68:

Question 16:

Evaluate:
$\sqrt[3]{64 \times 729}$

Answer:

$\sqrt[3]{64 \times 729}$
$\sqrt[3]{64 \times 729}$ = $\sqrt[3]{64} \times \sqrt[3]{729}$
= $\sqrt[3]{4 \times 4 \times 4}$ $\times \sqrt[3]{(3 \times 3 \times 3) \times (3 \times 3 \times 3)}$
= $\sqrt[3]{4 \times 4 \times 4}$ $\times \sqrt[3]{(9 \times 9 \times 9)}$
$\sqrt[3]{64 \times 729}$ = $(4) \times (9)$ = 36

Page No 68:

Question 17:

Evaluate:
$\sqrt[3]{\frac{729}{1000}}$

Answer:

$\sqrt[3]{\frac{729}{1000}}$

On factorisation:
$\sqrt[3]{\frac{729}{1000}}$ = $\frac{\sqrt[3]{(3 \times 3 \times 3) \times (3 \times 3 \times 3)}}{\sqrt[3]{(2 \times 2 \times 2) \times (5 \times 5 \times 5)}}$ = $\frac{\sqrt[3]{9 \times 9 \times 9}}{\sqrt[3]{10 \times 10 \times 10}}$
$\sqrt[3]{\frac{729}{1000}}$ = $\frac{9}{10}$

Page No 68:

Question 18:

Evaluate:
$\sqrt[3]{\frac{- 512}{343}}$

Answer:

$\sqrt[3]{\frac{- 512}{343}}$
By factorisation:

$\sqrt[3]{\frac{512}{343}}$ = $\frac{\sqrt[3]{8 \times 8 \times 8}}{\sqrt[3]{7 \times 7 \times 7}}$
$\sqrt[3]{\frac{- 512}{343}}$ = $\frac{- 8}{7}$

Page No 68:

Question 1:

Tick (✓) the correct answer
Which of the following numbers is a perfect cube?
(a) 141
(b) 294
(c) 216
(d) 496

Answer:

(a)
141 is not a perfect cube.

(b)
294 is not a perfect cube.

(c) (✓)
216 is a perfect cube.
216 = $(2 \times 2 \times 2) \times (3 \times 3 \times 3) = (2^{3}) \times (3^{3}) = 6^{3}$

(d)
496 is not a perfect cube.

Page No 68:

Question 2:

Tick (✓) the correct answer
Which of the following numbers is a perfect cube?
(a) 1152
(b) 1331
(c) 2016
(d) 739

Answer:

(a)
1152 = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 = {(2)}^{3} \times {(2)}^{3} \times (2 \times 3 \times 3)$ .
Hence, 1152 is not a perfect cube.

(b) (✓)
1331 = $11 \times 11 \times 11 = {(11)}^{3}$
Hence, 1331 is a perfect cube.

(c)
2016 = $2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 = {(2)}^{3} \times 2 \times 2 \times 3 \times 3 \times 7$
Hence, 2016 is not a perfect cube.

(d)
739 is not a perfect cube.

Page No 68:

Question 3:

Tick (✓) the correct answer
$\sqrt[3]{512} = ?$
(a) 6
(b) 7
(c) 8
(d) 9

Answer:

(c) 8

$\sqrt[3]{512}$ = $\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2} = \sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)}$
$\sqrt[3]{512}$ = $\sqrt[3]{{(2)}^{3} \times {(2)}^{3} \times {(2)}^{3}} = 8$

Hence, the cube root of 512 is 8.

Page No 68:

Question 4:

Tick (✓) the correct answer
$\sqrt[3]{125 \times 64} = ?$
(a) 100
(b) 40
(c) 20
(d) 30

Answer:

(c) 20

$\sqrt[3]{125 \times 64} = \sqrt[3]{125} \times \sqrt[3]{64} = \sqrt[3]{5 \times 5 \times 5} \times \sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2}$
$\sqrt[3]{125 \times 64} = \sqrt[3]{{(5)}^{3}} \times \sqrt[3]{{(2)}^{3} \times {(2)}^{3}} = \sqrt[3]{{(5)}^{3}} \times \sqrt[3]{{(4)}^{3}}$
$\sqrt[3]{125 \times 64} = 5 \times 4 = 20$

Hence, the cube root of $\sqrt[3]{125 \times 64}$ is 20.

Page No 68:

Question 5:

Tick (✓) the correct answer
$\sqrt[3]{\frac{64}{343}} = ?$
(a) $\frac{4}{9}$
(b) $\frac{4}{7}$
(c) $\frac{8}{7}$
(d) $\frac{8}{21}$

Answer:

(b) $\frac{4}{7}$
$\sqrt[3]{\frac{64}{343}}$ = $\frac{\sqrt[3]{64}}{\sqrt[3]{343}} = \frac{\sqrt[3]{4 \times 4 \times 4}}{\sqrt[3]{7 \times 7 \times 7}} = \frac{\sqrt[3]{{(4)}^{3}}}{\sqrt[3]{{(7)}^{3}}}$
$\sqrt[3]{\frac{64}{343}}$ = $\frac{4}{7}$
∴ $\sqrt[3]{\frac{64}{343}}$ = $\frac{4}{7}$

Page No 68:

Question 6:

Tick (✓) the correct answer
$\sqrt[3]{\frac{- 512}{729}} = ?$
(a) $\frac{- 7}{9}$
(b) $\frac{- 8}{9}$
(c) $\frac{7}{9}$
(d) $\frac{8}{9}$

Answer:

(b) $\frac{- 8}{9}$
$\sqrt[3]{\frac{- 512}{729}}$ = $\frac{\sqrt[3]{- 512}}{\sqrt[3]{729}} = \frac{\sqrt[3]{(- 8) \times (- 8) \times (- 8)}}{\sqrt[3]{9 \times 9 \times 9}} = \frac{\sqrt[3]{{(- 8)}^{3}}}{\sqrt[3]{{(9)}^{3}}}$
$\sqrt[3]{\frac{- 512}{729}}$ = $\frac{- 8}{9}$
∴ $\sqrt[3]{\frac{- 512}{729}}$ = $\frac{- 8}{9}$

Page No 68:

Question 7:

Tick (✓) the correct answer
By what least number should 648 be multiplied to get a perfect cube?
(a) 3
(b) 6
(c) 9
(d) 8

Answer:

(c) 9

648 = $2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 = {(2)}^{3} \times {(3)}^{3} \times 3$
Therefore, to get a perfect cube, we need to multiply 648 by 9, i.e. $(3 \times 3)$ .

Page No 68:

Question 8:

Tick (✓) the correct answer
By what least number should 1536 be divided to get a perfect cube?
(a) 3
(b) 4
(c) 6
(d) 8

Answer:

(a) 3

1536 = $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 = {(2)}^{3} \times {(2)}^{3} \times {(2)}^{3} \times 3$
Therefore, to get a perfect cube, we need to divide 1536 by 3.

Page No 68:

Question 9:

Tick (✓) the correct answer
${(1 \frac{3}{10})}^{3} = ?$
(a) $1 \frac{27}{1000}$
(b) $2 \frac{27}{1000}$
(c) $2 \frac{197}{1000}$
(d) none of these

Answer:

(c) $2 \frac{197}{1000}$
${(1 \frac{3}{10})}^{3} = {(\frac{13}{10})}^{3} = \frac{{(13)}^{3}}{{(10)}^{3}} = \frac{(13 \times 13 \times 13)}{(10 \times 10 \times 10)} {(1 \frac{3}{10})}^{3} = \frac{2197}{1000} = 2 \frac{197}{1000}$

∴ ${(1 \frac{3}{10})}^{3}$ = $2 \frac{197}{1000}$

Page No 68:

Question 10:

Tick (✓) the correct answer
(0.8)³ = ?
(a) 51.2
(b) 5.12
(c) 0.512
(d) none of these

Answer:

Page No 70:

Question 1:

Evaluate ${(1 \frac{2}{5})}^{3} .$

Answer:

${(1 \frac{2}{5})}^{3}$
${(1 \frac{2}{5})}^{3}$ = ${(\frac{7}{5})}^{3} = \frac{{(7)}^{3}}{{(5)}^{3}} = \frac{(7 \times 7 \times 7)}{(5 \times 5 \times 5)} = \frac{343}{125}$
∴ ${(1 \frac{2}{5})}^{3}$ = $\frac{343}{125}$

Page No 70:

Question 2:

Evaluate $\sqrt[3]{4096} .$

Answer:

$\sqrt[3]{4096}$

By prime factorisation method:

$\sqrt[3]{4096}$ = $\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2} = \sqrt[3]{{(2)}^{3} \times {(2)}^{3} \times {(2)}^{3} \times {(2)}^{3}}$
$\sqrt[3]{4096}$ = $(2) \times (2) \times (2) \times (2) = 16$ .

∴ $\sqrt[3]{4096}$ = $16$

Page No 70:

Question 3:

Evaluate $\sqrt[3]{216 \times 343} .$

Answer:

$\sqrt[3]{216 \times 343}$
By prime factorisation:

$\sqrt[3]{216 \times 343}$ = $\sqrt[3]{216} \times \sqrt[3]{343} = \sqrt[3]{2 \times 2 \times 2 \times 3 \times 3 \times 3} \times \sqrt[3]{7 \times 7 \times 7} = \sqrt[3]{{(2)}^{3} \times {(3)}^{3}} \times \sqrt[3]{{(7)}^{3}}$
$\sqrt[3]{216 \times 343}$ = $(2) \times (3) \times (7) = 42$

∴ $\sqrt[3]{216 \times 343}$ = $42$

Page No 70:

Question 4:

Evaluate $\sqrt[3]{\frac{- 64}{125}} .$

Answer:

$\sqrt[3]{\frac{- 64}{125}}$
By prime factorisation method:

$\sqrt[3]{\frac{- 64}{125}}$ = $\frac{\sqrt[3]{- 64}}{\sqrt[3]{125}} = \frac{\sqrt[3]{(- 4) \times (- 4) \times (- 4)}}{\sqrt[3]{5 \times 5 \times 5}} = \frac{\sqrt[3]{{(- 4)}^{3}}}{\sqrt[3]{{(5)}^{3}}}$
$\sqrt[3]{\frac{- 64}{125}}$ = $\frac{- 4}{5}$
∴ $\sqrt[3]{\frac{- 64}{125}}$ = $\frac{- 4}{5}$

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Question 5:

Mark (✓) against the correct answer
${(1 \frac{3}{4})}^{3} = ?$
(a) $1 \frac{27}{64}$
(b) $2 \frac{27}{64}$
(c) $5 \frac{23}{64}$
(d) none of these

Answer:

(c) $5 \frac{23}{64}$
${(1 \frac{3}{4})}^{3}$ = ${(\frac{7}{4})}^{3} = \frac{{(7)}^{3}}{{(4)}^{3}} = \frac{7 \times 7 \times 7}{4 \times 4 \times 4} = \frac{343}{64}$
${(1 \frac{3}{4})}^{3}$ = $\frac{343}{64}$ = $5 \frac{23}{64}$
∴ ${(1 \frac{3}{4})}^{3}$ = $5 \frac{23}{64}$

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Question 6:

Mark (✓) against the correct answer
Which of the following numbers is a perfect cube?
(a) 121
(b) 169
(c) 196
(d) 216

Answer:

(d) 216

$121 = 11 \times 11 169 = 13 \times 13 196 = 7 \times 7 \times 2 \times 2$

216 = $2 \times 2 \times 2 \times 3 \times 3 \times 3 = {(2)}^{3} \times {(3)}^{3} = {(6)}^{3}$

216 = 6³
Hence, 216 is a perfect cube.

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Question 7:

Mark (✓) against the correct answer
$\sqrt[3]{216 \times 64} = ?$
(a) 64
(b) 32
(c) 24
(d) 36

Answer:

(c) 24

$\sqrt[3]{216 \times 64}$ = $\sqrt[3]{216} \times \sqrt[3]{64} = \sqrt[3]{2 \times 2 \times 2 \times 3 \times 3 \times 3} \times \sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2}$
$\sqrt[3]{216 \times 64}$ = $\sqrt[3]{{(2)}^{3} \times {(3)}^{3}} \times \sqrt[3]{{(2)}^{3} \times {(2)}^{3}} = \sqrt[3]{{(6)}^{3}} \times \sqrt[3]{{(4)}^{3}}$
$\sqrt[3]{216 \times 64}$ = $6 \times 4 = 24$
∴ $\sqrt[3]{216 \times 64}$ = 24

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Question 8:

Mark (✓) against the correct answer
$\sqrt[3]{\frac{- 343}{729}} = ?$
(a) $\frac{7}{9}$
(b) $\frac{- 7}{9}$
(c) $\frac{- 9}{7}$
(d) $\frac{9}{7}$

Answer:

(b) $\frac{- 7}{9}$
By prime factorisation:
$\sqrt[3]{\frac{- 343}{729}}$ = $\frac{\sqrt[3]{- 343}}{\sqrt[3]{729}} = \frac{\sqrt[3]{(- 7) \times (- 7) \times (- 7)}}{\sqrt[3]{3 \times 3 \times 3 \times 3 \times 3 \times 3}} = \frac{\sqrt[3]{{(- 7)}^{3}}}{\sqrt[3]{{(3)}^{3} \times {(3)}^{3}}}$
$\sqrt[3]{\frac{- 343}{729}}$ = $\frac{\sqrt[3]{{(- 7)}^{3}}}{\sqrt[3]{{(9)}^{3}}}$ = $\frac{- 7}{9}$

∴ $\sqrt[3]{\frac{- 343}{729}}$ = $\frac{- 7}{9}$

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Question 9:

Mark (✓) against the correct answer
By what least number should 324 be multiplied to get a perfect cube?
(a) 12
(b) 14
(c) 16
(d) 18

Answer:

(d) 18

324 = $2 \times 2 \times 3 \times 3 \times 3 \times 3 = 2 \times 2 \times 3 \times {(3)}^{3}$

Therefore, to show that the given number is the product of three triplets, we need to multiply 324 by $(2 \times 3 \times 3)$ .
In other words, we need to multiply 324 by 18 to make it a perfect cube.

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Question 10:

Mark (✓) against the correct answer
$\frac{\sqrt[3]{128}}{\sqrt[3]{250}} = ?$
(a) $\frac{3}{5}$
(b) $\frac{4}{5}$
(c) $\frac{2}{5}$
(d) none of these

Answer:

(b) $\frac{4}{5}$

Resolving the numerator and the denominator into prime factors:

$\frac{\sqrt[3]{128}}{\sqrt[3]{250}} = \sqrt[3]{\frac{128}{250}} = \sqrt[3]{\frac{2 \times 8 \times 8}{2 \times 5 \times 5 \times 5}} = \sqrt[3]{\frac{2 \times 8 \times 8}{2 \times 5 \times 5 \times 5}} = \sqrt[3]{\frac{8 \times 8}{5 \times 5 \times 5}} = \sqrt[3]{\frac{{(2)}^{3} \times {(2)}^{3}}{{(5)}^{3}}} = \frac{2 \times 2}{5} = \frac{4}{5}$

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Question 11:

Mark (✓) against the correct answer
Which of the following is a cube of an odd number?
(a) 216
(b) 512
(c) 343
(d) 1000

Answer:

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Question 12:

Fill in the blanks.
(i) $\sqrt[3]{a b} = (\sqrt[3]{a}) \times (. . . . . . . . .) .$
(ii) $\sqrt[3]{\frac{a}{b}} = . . . . . . . . .$
(iii) $\sqrt[3]{- x} = . . . . . . . . .$
(iv) (0.5)³ = .........

Answer:

(i) $\sqrt[3]{b}$

$\sqrt[3]{a b} = (\sqrt[3]{a}) \times (\sqrt[3]{b})$

(ii) $\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$

$\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$

(iii) $- \sqrt[3]{x}$

$\sqrt[3]{- x} = - \sqrt[3]{x}$

(iv) 0.125

${(0.5)}^{3} = (0.5) \times (0.5) \times (0.5) = 0.125$

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