Question

Show that:

\[\frac{\sqrt[3]{729}}{\sqrt[3]{1000}} = \sqrt[3]{\frac{729}{1000}}\]

Answer 1

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.