Question

Show that:
(i) $\frac{\sqrt[3]{729}}{\sqrt[3]{1000}}=\sqrt[3]{\frac{729}{1000}}$
(ii) $\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}=\sqrt[3]{\frac{-512}{343}}$

Answer 1

(i)
LHS = $\frac{\sqrt[3]{729}}{\sqrt[3]{1000}}=\frac{\sqrt[3]{9\times 9\times 9}}{\sqrt[3]{10\times 10\times 10}}=\frac{9}{10}$

RHS = $\sqrt[3]{\frac{729}{1000}}=\sqrt[3]{\frac{9\times 9\times 9}{10\times 10\times 10}}=\sqrt[3]{\frac{9}{10}\times \frac{9}{10}\times \frac{9}{10}}=\sqrt[3]{{\left(\frac{9}{10}\right)}^3}=\frac{9}{10}$

Because LHS is equal to RHS, the equation is true.

(ii)
LHS = $\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}=\frac{-\sqrt[3]{512}}{\sqrt[3]{343}}=\frac{-\sqrt[3]{\left\{2\times 2\times 2\right\}\times \left\{2\times 2\times 2\right\}\times \left\{2\times 2\times 2\right\}}}{\sqrt[3]{7\times 7\times 7}}=\frac{-\left(2\times 2\times 2\right)}{7}=\frac{-8}{7}$

RHS =
$$\begin{gathered} \sqrt[3]{\frac{-512}{343}} \\ =\sqrt[3]{\frac{\left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)}{7\times 7\times 7}} \\ =\sqrt[3]{\frac{\left(-2\right)\times \left(-2\right)\times \left(-2\right)}{7}\times \frac{\left(-2\right)\times \left(-2\right)\times \left(-2\right)}{7}\times \frac{\left(-2\right)\times \left(-2\right)\times \left(-2\right)}{7}} \\ =\sqrt[3]{{\left(\frac{-8}{7}\right)}^3} \\ =\frac{-8}{7} \end{gathered}$$

Because LHS is equal to RHS, the equation is true.