Simplify the Radical Expression (81x^-24y^-12)^(1/4)

${(81 x^{- 24} y^{- 12})}^{\frac{1}{4}}$

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

${(81 \frac{1}{x^{24}} y^{- 12})}^{\frac{1}{4}}$

Combine $81$ and $\frac{1}{x^{24}}$ .

${(\frac{81}{x^{24}} y^{- 12})}^{\frac{1}{4}}$

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

${(\frac{81}{x^{24}} \cdot \frac{1}{y^{12}})}^{\frac{1}{4}}$

Combine.

${(\frac{81 \cdot 1}{x^{24} y^{12}})}^{\frac{1}{4}}$

Multiply $81$ by $1$ .

${(\frac{81}{x^{24} y^{12}})}^{\frac{1}{4}}$

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

Apply the product rule to $\frac{81}{x^{24} y^{12}}$ .

$\frac{81^{\frac{1}{4}}}{{(x^{24} y^{12})}^{\frac{1}{4}}}$

Apply the product rule to $x^{24} y^{12}$ .

$\frac{81^{\frac{1}{4}}}{{(x^{24})}^{\frac{1}{4}} {(y^{12})}^{\frac{1}{4}}}$

Simplify the numerator.

Rewrite $81$ as $3^{4}$ .

$\frac{{(3^{4})}^{\frac{1}{4}}}{{(x^{24})}^{\frac{1}{4}} {(y^{12})}^{\frac{1}{4}}}$

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

$\frac{3^{4 (\frac{1}{4})}}{{(x^{24})}^{\frac{1}{4}} {(y^{12})}^{\frac{1}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3^{4 (\frac{1}{4})}}{{(x^{24})}^{\frac{1}{4}} {(y^{12})}^{\frac{1}{4}}}$

Rewrite the expression.

$\frac{3^{1}}{{(x^{24})}^{\frac{1}{4}} {(y^{12})}^{\frac{1}{4}}}$

Evaluate the exponent.

$\frac{3}{{(x^{24})}^{\frac{1}{4}} {(y^{12})}^{\frac{1}{4}}}$

Simplify the denominator.

Multiply the exponents in ${(x^{24})}^{\frac{1}{4}}$ .

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

$\frac{3}{x^{24 (\frac{1}{4})} {(y^{12})}^{\frac{1}{4}}}$

Cancel the common factor of $4$ .

Factor $4$ out of $24$ .

$\frac{3}{x^{4 (6) \frac{1}{4}} {(y^{12})}^{\frac{1}{4}}}$

Cancel the common factor.

$\frac{3}{x^{4 \cdot 6 \frac{1}{4}} {(y^{12})}^{\frac{1}{4}}}$

Rewrite the expression.

$\frac{3}{x^{6} {(y^{12})}^{\frac{1}{4}}}$

Multiply the exponents in ${(y^{12})}^{\frac{1}{4}}$ .

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

$\frac{3}{x^{6} y^{12 (\frac{1}{4})}}$

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

$\frac{3}{x^{6} y^{4 (3) \frac{1}{4}}}$

Cancel the common factor.

$\frac{3}{x^{6} y^{4 \cdot 3 \frac{1}{4}}}$

Rewrite the expression.

$\frac{3}{x^{6} y^{3}}$

Simplify the Radical Expression (81x^-24y^-12)^(1/4)

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