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}

Answer:

(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 )  

Answer:

(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.

Answer:

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.

Answer:

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

Answer:

(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} 

Answer:

(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