Simplify (x^9y^6)^(-1/3)

${(x^{9} y^{6})}^{- \frac{1}{3}}$

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

$\frac{1}{{(x^{9} y^{6})}^{\frac{1}{3}}}$

Simplify the denominator.

Apply the product rule to $x^{9} y^{6}$ .

$\frac{1}{{(x^{9})}^{\frac{1}{3}} {(y^{6})}^{\frac{1}{3}}}$

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

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

$\frac{1}{x^{9 (\frac{1}{3})} {(y^{6})}^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the exponents in ${(y^{6})}^{\frac{1}{3}}$ .

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

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

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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