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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply the exponents in ${(y^{\frac{2}{3}})}^{6}$ .

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

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

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $2$ .

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

Step 6

Multiply the exponents in ${(x^{\frac{1}{2}})}^{6}$ .

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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