\displaystyle \textbf{Question 1: }\text{Find the cubes of:}
\displaystyle \text{(i) }16\qquad\text{(ii) }30\qquad\text{(iii) }1.2\qquad \text{(iv) }0.7
\displaystyle \text{(v) }0.06\qquad\text{(vi) }\frac{2}{5}\qquad \text{(vii) }\frac{1}{9}\qquad\text{(viii) }1\frac{3}{5}

\displaystyle \text{Answer:}
\displaystyle \text{(i) }16^3=16\times16\times16=4096.

\displaystyle \text{(ii) }30^3=30\times30\times30=27000.

\displaystyle \text{(iii) }(1.2)^3=1.2\times1.2\times1.2=1.728.

\displaystyle \text{(iv) }(0.7)^3=0.7\times0.7\times0.7=0.343.

\displaystyle \text{(v) }(0.06)^3=0.06\times0.06\times0.06=0.000216.

\displaystyle \text{(vi) }\left(\frac{2}{5}\right)^3 =\frac{2^3}{5^3}=\frac{8}{125}.

\displaystyle \text{(vii) }\left(\frac{1}{9}\right)^3 =\frac{1^3}{9^3}=\frac{1}{729}.

\displaystyle \text{(viii) }\left(1\frac{3}{5}\right)^3 =\left(\frac{8}{5}\right)^3=\frac{512}{125}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Test whether the given number is a perfect cube or not:}
\displaystyle \text{(i) }729\qquad\text{(ii) }3380\qquad \text{(iii) }10584\qquad\text{(iv) }13824

\displaystyle \text{Answer:}
\displaystyle \text{(i) }729=3^6=(3^2)^3=9^3.
\displaystyle \therefore 729\text{ is a perfect cube}.

\displaystyle \text{(ii) }3380=2^2\times5\times13^2.
\displaystyle \text{The prime factors cannot be grouped completely in triplets}.
\displaystyle \therefore 3380\text{ is not a perfect cube}.

\displaystyle \text{(iii) }10584=2^3\times3^3\times7^2.
\displaystyle \text{The prime factors cannot be grouped completely in triplets}.
\displaystyle \therefore 10584\text{ is not a perfect cube}.

\displaystyle \text{(iv) }13824=2^9\times3^3=(2^3\times3)^3=24^3.
\displaystyle \therefore 13824\text{ is a perfect cube}.
\displaystyle \\

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

\displaystyle \text{Answer:}
\displaystyle 11979=3\times3\times11\times11\times11=3^2\times11^3.
\displaystyle \text{To make a perfect cube, the prime factors must occur in groups of three}.
\displaystyle \therefore \text{The number must be multiplied by }3.
\displaystyle 11979\times3=35937=3^3\times11^3=33^3.
\displaystyle \therefore \text{The smallest number required is }3.
\displaystyle \\

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

\displaystyle \text{Answer:}
\displaystyle 8788=2\times2\times13\times13\times13=2^2\times13^3.
\displaystyle \text{To make a perfect cube, the prime factors must occur in groups of three}.
\displaystyle \therefore \text{The number must be divided by }2^2=4.
\displaystyle 8788\div4=2197=13^3.
\displaystyle \therefore \text{The smallest number required is }4.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Find the cube root of:}
\displaystyle \text{(i) }1728\qquad\text{(ii) }5832\qquad \text{(iii) }10648\qquad\text{(iv) }35937
\displaystyle \text{(v) }\frac{216}{2197}\qquad \text{(vi) }4\frac{508}{1331}\qquad\text{(vii) }42.875
\displaystyle \text{(viii) }0.000110592

\displaystyle \text{Answer:}
\displaystyle \text{(i) }1728=2^6\times3^3=(2^2\times3)^3=12^3.
\displaystyle \therefore \sqrt[3]{1728}=12.

\displaystyle \text{(ii) }5832=2^3\times3^6=(2\times3^2)^3=18^3.
\displaystyle \therefore \sqrt[3]{5832}=18.

\displaystyle \text{(iii) }10648=2^3\times11^3=(2\times11)^3=22^3.
\displaystyle \therefore \sqrt[3]{10648}=22.

\displaystyle \text{(iv) }35937=3^3\times11^3=(3\times11)^3=33^3.
\displaystyle \therefore \sqrt[3]{35937}=33.

\displaystyle \text{(v) }\frac{216}{2197} =\frac{6\times6\times6}{13\times13\times13} =\frac{6^3}{13^3}.
\displaystyle \therefore \sqrt[3]{\frac{216}{2197}}=\frac{6}{13}.

\displaystyle \text{(vi) }4\frac{508}{1331} =\frac{5832}{1331}=\frac{18^3}{11^3}.
\displaystyle \therefore \sqrt[3]{4\frac{508}{1331}}=\frac{18}{11}.

\displaystyle \text{(vii) }42.875=\frac{42875}{1000} =\frac{35^3}{10^3}.
\displaystyle \therefore \sqrt[3]{42.875}=\frac{35}{10}=3.5.

\displaystyle \text{(viii) }0.000110592 =\frac{110592}{1000000000}=\frac{48^3}{1000^3}.
\displaystyle \therefore \sqrt[3]{0.000110592} =\frac{48}{1000}=0.048.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Evaluate:}
\displaystyle \text{(i) }\sqrt[3]{1352}\times\sqrt[3]{1625}\qquad \text{(ii) }\sqrt[3]{\frac{51.2}{0.4096}}
\displaystyle \text{(iii) }\sqrt[3]{\sqrt{0.000064}}\qquad \text{(iv) }\sqrt[3]{\sqrt{0.004096}}+\sqrt[3]{0.729}

\displaystyle \text{Answer:}
\displaystyle \text{(i) }\sqrt[3]{1352}\times\sqrt[3]{1625} =\sqrt[3]{1352\times1625}
\displaystyle =\sqrt[3]{(2^3\times13^2)(5^3\times13)} =\sqrt[3]{2^3\times5^3\times13^3}
\displaystyle =2\times5\times13=130.

\displaystyle \text{(ii) }\sqrt[3]{\frac{51.2}{0.4096}} =\sqrt[3]{125}=5.

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

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

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