Calculation:

According to quotient property of square root $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

And the product property of square root states $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$.

Given expression is $\sqrt[3]{\frac{7}{8{x}^{3}}}$

This can be written as $\sqrt[3]{\frac{7}{8{x}^{3}}}=\sqrt[3]{\frac{7}{2\times 2\times 2\times {x}^{3}}}$

$\sqrt[3]{\frac{7}{8{x}^{3}}}=\sqrt[3]{\frac{7}{2\times 2\times 2\times {x}^{3}}}$

Applying the quotient rule,

$\sqrt[3]{\frac{7}{8{x}^{3}}}=\frac{\sqrt[3]{7}}{\sqrt[3]{2\times 2\times 2\times {x}^{3}}}$

Applying the product rule for Denominator,

$\sqrt[3]{\frac{7}{8{x}^{3}}}=\frac{\sqrt[3]{7}}{\sqrt[3]{(2{)}^{3}\times \sqrt[3]{{x}^{3}}}}$

$\sqrt[3]{\frac{7}{8{x}^{3}}}=\frac{\sqrt[3]{7}}{2x}$