Simplify the Radical Expression (32x^-30y^-15)^(-4/5)

${(32 x^{- 30} y^{- 15})}^{- \frac{4}{5}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(32 \frac{1}{x^{30}} y^{- 15})}^{- \frac{4}{5}}$

Combine $32$ and $\frac{1}{x^{30}}$ .

${(\frac{32}{x^{30}} y^{- 15})}^{- \frac{4}{5}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{32}{x^{30}} \cdot \frac{1}{y^{15}})}^{- \frac{4}{5}}$

Combine.

${(\frac{32 \cdot 1}{x^{30} y^{15}})}^{- \frac{4}{5}}$

Multiply $32$ by $1$ .

${(\frac{32}{x^{30} y^{15}})}^{- \frac{4}{5}}$

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{x^{30} y^{15}}{32})}^{\frac{4}{5}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{x^{30} y^{15}}{32}$ .

$\frac{{(x^{30} y^{15})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Apply the product rule to $x^{30} y^{15}$ .

$\frac{{(x^{30})}^{\frac{4}{5}} {(y^{15})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Simplify the numerator.

Multiply the exponents in ${(x^{30})}^{\frac{4}{5}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{30 (\frac{4}{5})} {(y^{15})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Cancel the common factor of $5$ .

Factor $5$ out of $30$ .

$\frac{x^{5 (6) \frac{4}{5}} {(y^{15})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Cancel the common factor.

$\frac{x^{5 \cdot 6 \frac{4}{5}} {(y^{15})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Rewrite the expression.

$\frac{x^{6 \cdot 4} {(y^{15})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Multiply $6$ by $4$ .

$\frac{x^{24} {(y^{15})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Multiply the exponents in ${(y^{15})}^{\frac{4}{5}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{24} y^{15 (\frac{4}{5})}}{32^{\frac{4}{5}}}$

Cancel the common factor of $5$ .

Factor $5$ out of $15$ .

$\frac{x^{24} y^{5 (3) \frac{4}{5}}}{32^{\frac{4}{5}}}$

Cancel the common factor.

$\frac{x^{24} y^{5 \cdot 3 \frac{4}{5}}}{32^{\frac{4}{5}}}$

Rewrite the expression.

$\frac{x^{24} y^{3 \cdot 4}}{32^{\frac{4}{5}}}$

Multiply $3$ by $4$ .

$\frac{x^{24} y^{12}}{32^{\frac{4}{5}}}$

Simplify the denominator.

Rewrite $32$ as $2^{5}$ .

$\frac{x^{24} y^{12}}{{(2^{5})}^{\frac{4}{5}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{24} y^{12}}{2^{5 (\frac{4}{5})}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{x^{24} y^{12}}{2^{5 (\frac{4}{5})}}$

Rewrite the expression.

$\frac{x^{24} y^{12}}{2^{4}}$

Raise $2$ to the power of $4$ .

$\frac{x^{24} y^{12}}{16}$

Simplify the Radical Expression (32x^-30y^-15)^(-4/5)

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