Simplify the Radical Expression (64x^-21y^9)^(1/3)

${(64 x^{- 21} y^{9})}^{\frac{1}{3}}$

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

${(64 \frac{1}{x^{21}} y^{9})}^{\frac{1}{3}}$

Combine $64$ and $\frac{1}{x^{21}}$ .

${(\frac{64}{x^{21}} y^{9})}^{\frac{1}{3}}$

Combine $\frac{64}{x^{21}}$ and $y^{9}$ .

${(\frac{64 y^{9}}{x^{21}})}^{\frac{1}{3}}$

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

Apply the product rule to $\frac{64 y^{9}}{x^{21}}$ .

$\frac{{(64 y^{9})}^{\frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

Apply the product rule to $64 y^{9}$ .

$\frac{64^{\frac{1}{3}} {(y^{9})}^{\frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

Simplify the numerator.

Rewrite $64$ as $4^{3}$ .

$\frac{{(4^{3})}^{\frac{1}{3}} {(y^{9})}^{\frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

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

$\frac{4^{3 (\frac{1}{3})} {(y^{9})}^{\frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{4^{3 (\frac{1}{3})} {(y^{9})}^{\frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

Rewrite the expression.

$\frac{4^{1} {(y^{9})}^{\frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

Evaluate the exponent.

$\frac{4 {(y^{9})}^{\frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

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

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

$\frac{4 y^{9 (\frac{1}{3})}}{{(x^{21})}^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$\frac{4 y^{3 (3) \frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

Cancel the common factor.

$\frac{4 y^{3 \cdot 3 \frac{1}{3}}}{{(x^{21})}^{\frac{1}{3}}}$

Rewrite the expression.

$\frac{4 y^{3}}{{(x^{21})}^{\frac{1}{3}}}$

Multiply the exponents in ${(x^{21})}^{\frac{1}{3}}$ .

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

$\frac{4 y^{3}}{x^{21 (\frac{1}{3})}}$

Cancel the common factor of $3$ .

Factor $3$ out of $21$ .

$\frac{4 y^{3}}{x^{3 (7) \frac{1}{3}}}$

Cancel the common factor.

$\frac{4 y^{3}}{x^{3 \cdot 7 \frac{1}{3}}}$

Rewrite the expression.

$\frac{4 y^{3}}{x^{7}}$