Show that: [3] -512 [3] 343 = [3] -512 343

Question

Show that:
$\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}=\sqrt[3]{\frac{-512}{343}}$

✓

Answer

$\text{LHS}=\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}$
$=\frac{-\sqrt[3]{512}}{\sqrt[3]{343}}$
$=\frac{\sqrt[3]{\{2\times2\times2\} \times\{2\times2\times2\}\times\{2\times2\times2\}}}{\sqrt[3]{7\times7\times7}}$
$=\frac{-(2\times2\times2)}{7}$
$=\frac{-8}{7}$
$\text{RHS}=\sqrt[3]\frac{{-512}}{{343}}$
$=\sqrt[3]\frac{{(-2)\times(-2)\times(-2)\times(-2)\times(-2)\times(-2)\times(-2)\times(-2)\times(-2)}}{{7\times7\times7}}$
$=\sqrt[3]{\frac{(-2)\times(-2)\times(-2)}{7}\times\frac{(-2)\times(-2)\times(-2)}{7}\times\frac{(-2)\times(-2)\times(-2)}{7}}$
$=\sqrt[3]{(\frac{-8}{7})^3}$
$=\frac{-8}{7}$
Because LHS is equal to RHS, the equation is true.

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