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}   

Answer:

(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}   

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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 )  

Answer:

(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)

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

Answer:

\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.

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

Answer:

\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.

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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

Answer:

(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

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Question 6: Evaluate:

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

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

Answer:

(i) \displaystyle  \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

\displaystyle  \text{ 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) \displaystyle  \sqrt[3]{\sqrt{0.000064}} = \sqrt[3]{0.008} = 0.2

(iv) \displaystyle  \sqrt[3]{\sqrt{0.004096}} + \sqrt[3]{0.729} = \sqrt[3]{0.064} +0.9=0.4+0.9=1.3