Rewrite \(4\) as \({2}^{2}\).
\[\begin{aligned}&\frac{{125}^{-\frac{2}{3}}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-{(\sqrt[3]{{2}^{2}})}^{0}\times \frac{{(-2)}^{5}}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+{(\frac{81}{16})}^{-\frac{1}{4}}\end{aligned}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\begin{aligned}&\frac{{125}^{-\frac{2}{3}}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-{({2}^{\frac{2}{3}})}^{0}\times \frac{{(-2)}^{5}}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+{(\frac{81}{16})}^{-\frac{1}{4}}\end{aligned}\]
Calculate.
\[\begin{aligned}&\frac{{5}^{-2}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-{({2}^{\frac{2}{3}})}^{0}\times \frac{{(-2)}^{5}}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+{(\frac{81}{16})}^{-\frac{1}{4}}\end{aligned}\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\begin{aligned}&\frac{\frac{1}{{5}^{2}}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-{({2}^{\frac{2}{3}})}^{0}\times \frac{{(-2)}^{5}}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+{(\frac{81}{16})}^{-\frac{1}{4}}\end{aligned}\]
Simplify \({5}^{2}\) to \(25\).
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-{({2}^{\frac{2}{3}})}^{0}\times \frac{{(-2)}^{5}}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+{(\frac{81}{16})}^{-\frac{1}{4}}\end{aligned}\]
Use Rule of Zero: \({x}^{0}=1\).
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-1\times \frac{{(-2)}^{5}}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+{(\frac{81}{16})}^{-\frac{1}{4}}\end{aligned}\]
Since the power of 5 is odd, the result will be negative.
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-1\times \frac{-{2}^{5}}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+{(\frac{81}{16})}^{-\frac{1}{4}}\end{aligned}\]
Simplify \({2}^{5}\) to \(32\).
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-1\times \frac{-32}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+{(\frac{81}{16})}^{-\frac{1}{4}}\end{aligned}\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-1\times \frac{-32}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+\frac{1}{\sqrt[4]{\frac{81}{16}}}\end{aligned}\]
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-1\times \frac{-32}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+\frac{1}{\frac{\sqrt[4]{81}}{\sqrt[4]{16}}}\end{aligned}\]
Calculate.
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-1\times \frac{-32}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+\frac{1}{\frac{3}{\sqrt[4]{16}}}\end{aligned}\]
Calculate.
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-1\times \frac{-32}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+\frac{1}{\frac{3}{2}}\end{aligned}\]
Invert and multiply.
\[\begin{aligned}&\frac{\frac{1}{25}}{{8}^{\frac{2}{3}}}\\&{243}^{\frac{2}{5}}+{32}^{-\frac{2}{5}}\\&{(-3)}^{4}-1\times \frac{-32}{{64}^{\frac{2}{3}}}\\&{27}^{\frac{2}{3}}+\frac{2}{3}\end{aligned}\]
(1/25)/8^(2/3);243^(2/5)+32^(-2/5);(-3)^4-1*(-32)/64^(2/3);27^(2/3)+2/3
