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Solution:

(i)

$L H S=\frac{\sqrt[3]{729}}{\sqrt[3]{1000}}=\frac{\sqrt[3]{9 \times 9 \times 9}}{\sqrt[3]{10 \times 10 \times 10}}=\frac{9}{10}$

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

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

(ii)

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

RHS=

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

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

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

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

$=\frac{-8}{7}$

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