Question

Find the cube root of the following rational number  \[\frac{686}{- 3456}\] .

Answer 1

Let us consider the following rational number: 

\[\frac{686}{- 3456}\] 

\[\sqrt[3]{\frac{686}{- 3456}}\]

\[= - \sqrt[3]{\frac{2 \times 7^3}{2^7 \times 3^3}}\]

 (686 and 3456 are not perfect cubes; therefore, we simplify it as \[\frac{686}{3456}\]  by prime factorisation.)

\[= - \sqrt[3]{\frac{7^3}{2^6 \times 3^3}}\]

\[= \frac{- \sqrt[3]{7^3}}{\sqrt[3]{2^6 \times 3^3}}\]

\[ = \frac{- 7}{\sqrt[3]{2^3 \times 2^3 \times 3^3}}\]

\[ = \frac{- 7}{2 \times 2 \times 3}\]

\[ = \frac{- 7}{12}\]

( ∵ \[\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\] )