Question 1: Find the cubes of:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question 3: Find the smallest number by which $\displaystyle 11979$ must be multiplied so that the product is a perfect cube. $\displaystyle 11979 = 3 \times 3 \times 11 \times 11 \times 11$

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

Question 4: Find the smallest number by which $\displaystyle 8788$ must be divided so that the quotient is a perfect cube. $\displaystyle 8788 = 2 \times 2 \times 13 \times 13 \times 13$

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

Question 5: Find the cube root of:

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

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

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

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

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

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

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

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

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

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

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

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

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

Question 6: Evaluate:

(i) $\displaystyle \sqrt{1352} \times \sqrt{1625 }$     (ii) $\displaystyle \sqrt{ \frac{51.2}{0.4096}}$

(iii) $\displaystyle \sqrt{\sqrt{0.000064}}$     (iv) $\displaystyle \sqrt{\sqrt{0.004096}} + \sqrt{0.729}$

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