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

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

Move $y^{- 9}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Multiply $y^{6}$ by $y^{9}$ by adding the exponents.

Move $y^{9}$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $9$ and $6$ .

${(\frac{x^{6}}{x^{- 3} y^{15}})}^{\frac{1}{3}}$

Move $x^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{x^{6} x^{3}}{y^{15}})}^{\frac{1}{3}}$

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{x^{6 + 3}}{y^{15}})}^{\frac{1}{3}}$

Add $6$ and $3$ .

${(\frac{x^{9}}{y^{15}})}^{\frac{1}{3}}$

Apply the product rule to $\frac{x^{9}}{y^{15}}$ .

$\frac{{(x^{9})}^{\frac{1}{3}}}{{(y^{15})}^{\frac{1}{3}}}$

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

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

$\frac{x^{9 (\frac{1}{3})}}{{(y^{15})}^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$\frac{x^{3 (3) \frac{1}{3}}}{{(y^{15})}^{\frac{1}{3}}}$

Cancel the common factor.

$\frac{x^{3 \cdot 3 \frac{1}{3}}}{{(y^{15})}^{\frac{1}{3}}}$

Rewrite the expression.

$\frac{x^{3}}{{(y^{15})}^{\frac{1}{3}}}$

Multiply the exponents in ${(y^{15})}^{\frac{1}{3}}$ .

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

$\frac{x^{3}}{y^{15 (\frac{1}{3})}}$

Cancel the common factor of $3$ .

Factor $3$ out of $15$ .

$\frac{x^{3}}{y^{3 (5) \frac{1}{3}}}$

Cancel the common factor.

$\frac{x^{3}}{y^{3 \cdot 5 \frac{1}{3}}}$

Rewrite the expression.

$\frac{x^{3}}{y^{5}}$