Simplify the Radical Expression (-27x^-9)^(1/3)

${(- 27 x^{- 9})}^{\frac{1}{3}}$

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

${(- 27 \frac{1}{x^{9}})}^{\frac{1}{3}}$

Combine $- 27$ and $\frac{1}{x^{9}}$ .

${(\frac{- 27}{x^{9}})}^{\frac{1}{3}}$

Move the negative in front of the fraction.

${(- \frac{27}{x^{9}})}^{\frac{1}{3}}$

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

Apply the product rule to $- \frac{27}{x^{9}}$ .

${(- 1)}^{\frac{1}{3}} {(\frac{27}{x^{9}})}^{\frac{1}{3}}$

Apply the product rule to $\frac{27}{x^{9}}$ .

${(- 1)}^{\frac{1}{3}} \frac{27^{\frac{1}{3}}}{{(x^{9})}^{\frac{1}{3}}}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

${({(- 1)}^{3})}^{\frac{1}{3}} \frac{27^{\frac{1}{3}}}{{(x^{9})}^{\frac{1}{3}}}$

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

${(- 1)}^{3 (\frac{1}{3})} \frac{27^{\frac{1}{3}}}{{(x^{9})}^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(- 1)}^{3 (\frac{1}{3})} \frac{27^{\frac{1}{3}}}{{(x^{9})}^{\frac{1}{3}}}$

Rewrite the expression.

${(- 1)}^{1} \frac{27^{\frac{1}{3}}}{{(x^{9})}^{\frac{1}{3}}}$

Evaluate the exponent.

$- \frac{27^{\frac{1}{3}}}{{(x^{9})}^{\frac{1}{3}}}$

Simplify the numerator.

Rewrite $27$ as $3^{3}$ .

$- \frac{{(3^{3})}^{\frac{1}{3}}}{{(x^{9})}^{\frac{1}{3}}}$

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

$- \frac{3^{3 (\frac{1}{3})}}{{(x^{9})}^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- \frac{3^{3 (\frac{1}{3})}}{{(x^{9})}^{\frac{1}{3}}}$

Rewrite the expression.

$- \frac{3^{1}}{{(x^{9})}^{\frac{1}{3}}}$

Evaluate the exponent.

$- \frac{3}{{(x^{9})}^{\frac{1}{3}}}$

Multiply the exponents in ${(x^{9})}^{\frac{1}{3}}$ .

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

$- \frac{3}{x^{9 (\frac{1}{3})}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$- \frac{3}{x^{3 (3) \frac{1}{3}}}$

Cancel the common factor.

$- \frac{3}{x^{3 \cdot 3 \frac{1}{3}}}$

Rewrite the expression.

$- \frac{3}{x^{3}}$