Simplify the Radical Expression (8x^3y^9)^(-2/3)

${(8 x^{3} y^{9})}^{- \frac{2}{3}}$

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

$\frac{1}{{(8 x^{3} y^{9})}^{\frac{2}{3}}}$

Simplify the denominator.

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

Apply the product rule to $8 x^{3} y^{9}$ .

$\frac{1}{{(8 x^{3})}^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}$

Apply the product rule to $8 x^{3}$ .

$\frac{1}{8^{\frac{2}{3}} {(x^{3})}^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}$

Rewrite $8$ as $2^{3}$ .

$\frac{1}{{(2^{3})}^{\frac{2}{3}} {(x^{3})}^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}$

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

$\frac{1}{2^{3 (\frac{2}{3})} {(x^{3})}^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{1}{2^{3 (\frac{2}{3})} {(x^{3})}^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{1}{2^{2} {(x^{3})}^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}$

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

$\frac{1}{4 {(x^{3})}^{\frac{2}{3}} {(y^{9})}^{\frac{2}{3}}}$

Multiply the exponents in ${(x^{3})}^{\frac{2}{3}}$ .

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

$\frac{1}{4 x^{3 (\frac{2}{3})} {(y^{9})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{1}{4 x^{3 (\frac{2}{3})} {(y^{9})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{1}{4 x^{2} {(y^{9})}^{\frac{2}{3}}}$

Multiply the exponents in ${(y^{9})}^{\frac{2}{3}}$ .

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

$\frac{1}{4 x^{2} y^{9 (\frac{2}{3})}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$\frac{1}{4 x^{2} y^{3 (3) \frac{2}{3}}}$

Cancel the common factor.

$\frac{1}{4 x^{2} y^{3 \cdot 3 \frac{2}{3}}}$

Rewrite the expression.

$\frac{1}{4 x^{2} y^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{1}{4 x^{2} y^{6}}$