1Cademy - Simplifying $$\sqrt[3]{\frac{16}{54}}$$ and $$\sqrt[4]{\frac{5}{80}}$$

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Simplifying $\sqrt{\frac{45}{80}}$

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Simplifying $\sqrt[3]{\frac{16}{54}}$ and $\sqrt[4]{\frac{5}{80}}$

Simplify higher-order roots whose radicands are fractions by first removing common factors from the numerator and denominator to reveal perfect powers.

ⓐ $\sqrt[3]{\frac{16}{54}}$ : Simplify inside the radical first. Factor to reveal common factors: $\sqrt[3]{\frac{2 \cdot 8}{2 \cdot 27}}$ . Remove the common factor of $2$ to get $\sqrt[3]{\frac{8}{27}}$ . The reduced fraction is a perfect cube fraction, since $\left(\frac{2}{3}\right)^3 = \frac{8}{27}$ . The result is $\frac{2}{3}$ .

ⓑ $\sqrt[4]{\frac{5}{80}}$ : Simplify inside the radical. Factor to reveal common factors: $\sqrt[4]{\frac{5 \cdot 1}{5 \cdot 16}}$ . Remove the common factor of $5$ to get $\sqrt[4]{\frac{1}{16}}$ . The reduced fraction is a perfect fourth power fraction, since $\left(\frac{1}{2}\right)^4 = \frac{1}{16}$ . The result is $\frac{1}{2}$ .