Question

Show that: 

\[\frac{\sqrt[3]{- 512}}{\sqrt[3]{343}} = \sqrt[3]{\frac{- 512}{343}}\]

Answer 1

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 = 

\[\sqrt[3]{\frac{- 512}{343}}\]

\[ = \sqrt[3]{\frac{\left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right)}{7 \times 7 \times 7}}\]

\[ = \sqrt[3]{\frac{\left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right)}{7} \times \frac{\left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right)}{7} \times \frac{\left( - 2 \right) \times \left( - 2 \right) \times \left( - 2 \right)}{7}}\]

\[ = \sqrt[3]{\left( \frac{- 8}{7} \right)^3}\]

\[ = \frac{- 8}{7}\]

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