Simplify (x^(2/3)y^(-1/6))^-12

${(x^{\frac{2}{3}} y^{- \frac{1}{6}})}^{- 12}$

Step 1

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

${(x^{\frac{2}{3}} \frac{1}{y^{\frac{1}{6}}})}^{- 12}$

Step 2

Combine $x^{\frac{2}{3}}$ and $\frac{1}{y^{\frac{1}{6}}}$ .

${(\frac{x^{\frac{2}{3}}}{y^{\frac{1}{6}}})}^{- 12}$

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{y^{\frac{1}{6}}}{x^{\frac{2}{3}}})}^{12}$

Step 4

Apply the product rule to $\frac{y^{\frac{1}{6}}}{x^{\frac{2}{3}}}$ .

$\frac{{(y^{\frac{1}{6}})}^{12}}{{(x^{\frac{2}{3}})}^{12}}$

Step 5

Multiply the exponents in ${(y^{\frac{1}{6}})}^{12}$ .

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

$\frac{y^{\frac{1}{6} \cdot 12}}{{(x^{\frac{2}{3}})}^{12}}$

Cancel the common factor of $6$ .

Factor $6$ out of $12$ .

$\frac{y^{\frac{1}{6} \cdot (6 (2))}}{{(x^{\frac{2}{3}})}^{12}}$

Cancel the common factor.

$\frac{y^{\frac{1}{6} \cdot (6 \cdot 2)}}{{(x^{\frac{2}{3}})}^{12}}$

Rewrite the expression.

$\frac{y^{2}}{{(x^{\frac{2}{3}})}^{12}}$

Step 6

Multiply the exponents in ${(x^{\frac{2}{3}})}^{12}$ .

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

$\frac{y^{2}}{x^{\frac{2}{3} \cdot 12}}$

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $4$ .

$\frac{y^{2}}{x^{8}}$