Question 1: Find the cubes of:

(i) $16$     (ii) $30$      (iii) $1.2$      (iv) $0.7$      (v) $0.06$      (vi) $\frac{2}{5}$      (vii) $\frac{1}{9}$      (viii) $1$ $\frac{3}{5}$

(i) $16^3 = 16 \times 16 \times 16 = 4096$

(ii) $30^3 = 30 \times 30 \times 30 = 27000$

(iii) $1.2^3 = 1.2 \times 1.2 \times 1.2 = 1.728$

(iv) $0.7^3 = 0.7 \times 0.7 \times 0.7 = 0.343$

(v) $0.06^3 = 0.06 \times 0.06 \times 0.06 = 0.000216$

(vi) $\Big($ ${\frac{2}{5}}$ $\Big)^3$ $=$ $\frac{8}{125}$

(vii) $\Big($ ${\frac{1}{9}}$ $\Big)^3$ $=$ $\frac{1}{729}$

(viii) $\Big($ ${\frac{3}{5}}$ $\Big)^3$ $=$ $\frac{512}{125}$

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Question 2: Test whether the given number is a perfect cube or not:

(i)  $729$       (ii)  $3380$      (iii) $10584$   (iv) $13824 )$

(i)  $729 = 7 \times 7 \times 7$ (hence a perfect cube)

(ii)  $3380 = 4 \times 5 \times 13 \times 13$ (hence not a perfect cube)

(iii) $10584 = 8 \times 3 \times 3 \times 3 \times 3 \times 7 \times 7$ (hence not a perfect cube)

(iv) $13824 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3)$ (hence a perfect cube)

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Question 3: Find the smallest number by which $11979$ must be multiplied so that the product is a perfect cube.

$11979 = 3 \times 3 \times 11 \times 11 \times 11$

Therefore multiply the number by $3$ for it to be a perfect cube.

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Question 4: Find the smallest number by which $8788$ must be divided so that the quotient is a perfect cube.

$8788 = 2 \times 2 \times 13 \times 13 \times 13$

Hence divide the number by $4$ so that the quotient is a perfect square.

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Question 5: Find the cube root of:

(i) $1728$     (ii) $5832$     (iii) $10648$      (iv) $35937$

(v) $\frac{216}{2197}$      (vi) $4$ $\frac{508}{1331}$       (vii) $42.875$      (viii) $0.000110592$

(i) $1728 = 23 \times 23 \times 33$ Therefore the cube root of $1728$ is $12$

(ii) $5832 = 23 \times 93$ Therefore the cube root of $5832$ is $18$

(iii) $10648 = 23 \times 113$ Therefore the cube root of $10648$ is $22$

(iv) $35937 = 33 \times 113$ Therefore the cube root of $35937$ is $33$

(v) $\frac{216}{2197}$ $=$ $\frac{6 \times 6 \times 6}{13 \times 13 \times 13}$ Therefore the cube root of $\frac{216}{2197}$ is $\frac{6}{13}$

(vi) $4$ $\frac{508}{1331}$ $=$ $\frac{5832}{1331}$ $=$ $\frac{2 \times 2 \times 2 \times 9 \times 9 \times 9}{11 \times 11 \times 11}$

Therefore the cube root of $4$ $\frac{508}{1331}$ $=$ $\frac{18}{11}$

(vii) $42.875 =$ $\frac{42875}{1000}$ $=$ $\frac{5 \times 5 \times 5 \times 7 \times 7 \times 7}{10 \times 10 \times 10}$

Therefore the cube root of $42.875$ is $3.5$

(viii) $0.000110592 =$ $\frac{110592}{1000000000}$ $=$ $\frac{4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 3 \times 3 \times 3}{1000 \times 1000 \times 1000}$

Therefore the cube root of $0.000110592$ is $0.048$

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Question 6: Evaluate:

(i) $\sqrt[3]{1352} \times \sqrt[3]{1625 }$     (ii) $\sqrt[3]{ \frac{51.2}{0.4096}}$

(iii) $\sqrt[3]{\sqrt{0.000064}}$     (iv) $\sqrt[3]{\sqrt{0.004096}} + \sqrt[3]{0.729}$

(i) $\sqrt[3]{1352} \times \sqrt[3]{1625 } = \sqrt[3]{2 \times 2 \times 2 \times 13 \times 13} \times \sqrt[3]{5 \times 5 \times 5 \times 13} =2 \times 5 \times 13=130$
(ii) $\sqrt[3]{ \frac{51.2}{0.4096}}$ $=$ $\sqrt[3]{\frac{512000}{4096}}$ $=$ $\sqrt[3]{ \frac{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 10 \times 10 \times 10}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}}$ $=5$
(iii) $\sqrt[3]{\sqrt{0.000064}} = \sqrt[3]{0.008} = 0.2$
(iv) $\sqrt[3]{\sqrt{0.004096}} + \sqrt[3]{0.729} = \sqrt[3]{0.064} +0.9=0.4+0.9=1.3$