Simplify the Radical Expression (32x^-30y^-35)^(-1/5)

${(32 x^{- 30} y^{- 35})}^{- \frac{1}{5}}$

Step 1

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

${(32 \frac{1}{x^{30}} y^{- 35})}^{- \frac{1}{5}}$

Step 2

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

${(\frac{32}{x^{30}} y^{- 35})}^{- \frac{1}{5}}$

Step 3

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

${(\frac{32}{x^{30}} \cdot \frac{1}{y^{35}})}^{- \frac{1}{5}}$

Step 4

Combine.

${(\frac{32 \cdot 1}{x^{30} y^{35}})}^{- \frac{1}{5}}$

Step 5

Multiply $32$ by $1$ .

${(\frac{32}{x^{30} y^{35}})}^{- \frac{1}{5}}$

Step 6

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

${(\frac{x^{30} y^{35}}{32})}^{\frac{1}{5}}$

Step 7

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

Apply the product rule to $\frac{x^{30} y^{35}}{32}$ .

$\frac{{(x^{30} y^{35})}^{\frac{1}{5}}}{32^{\frac{1}{5}}}$

Apply the product rule to $x^{30} y^{35}$ .

$\frac{{(x^{30})}^{\frac{1}{5}} {(y^{35})}^{\frac{1}{5}}}{32^{\frac{1}{5}}}$

Step 8

Simplify the numerator.

Multiply the exponents in ${(x^{30})}^{\frac{1}{5}}$ .

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

$\frac{x^{30 (\frac{1}{5})} {(y^{35})}^{\frac{1}{5}}}{32^{\frac{1}{5}}}$

Cancel the common factor of $5$ .

Factor $5$ out of $30$ .

$\frac{x^{5 (6) \frac{1}{5}} {(y^{35})}^{\frac{1}{5}}}{32^{\frac{1}{5}}}$

Cancel the common factor.

$\frac{x^{5 \cdot 6 \frac{1}{5}} {(y^{35})}^{\frac{1}{5}}}{32^{\frac{1}{5}}}$

Rewrite the expression.

$\frac{x^{6} {(y^{35})}^{\frac{1}{5}}}{32^{\frac{1}{5}}}$

Multiply the exponents in ${(y^{35})}^{\frac{1}{5}}$ .

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

$\frac{x^{6} y^{35 (\frac{1}{5})}}{32^{\frac{1}{5}}}$

Cancel the common factor of $5$ .

Factor $5$ out of $35$ .

$\frac{x^{6} y^{5 (7) \frac{1}{5}}}{32^{\frac{1}{5}}}$

Cancel the common factor.

$\frac{x^{6} y^{5 \cdot 7 \frac{1}{5}}}{32^{\frac{1}{5}}}$

Rewrite the expression.

$\frac{x^{6} y^{7}}{32^{\frac{1}{5}}}$

Step 9

Simplify the denominator.

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

$\frac{x^{6} y^{7}}{{(2^{5})}^{\frac{1}{5}}}$

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

$\frac{x^{6} y^{7}}{2^{5 (\frac{1}{5})}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{x^{6} y^{7}}{2^{5 (\frac{1}{5})}}$

Rewrite the expression.

$\frac{x^{6} y^{7}}{2^{1}}$

Evaluate the exponent.

$\frac{x^{6} y^{7}}{2}$

Simplify the Radical Expression (32x^-30y^-35)^(-1/5)

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