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

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

Step 1

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

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

Step 2

Simplify the denominator.

Multiply $x^{\frac{2}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

Move $x^{\frac{1}{3}}$ .

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $2$ .

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

Divide $3$ by $3$ .

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

Simplify $x^{1} y^{- \frac{1}{2}}$ .

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

Step 3

Move $y^{- \frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 4

Multiply $y$ by $y^{\frac{1}{2}}$ by adding the exponents.

Multiply $y$ by $y^{\frac{1}{2}}$ .

Raise $y$ to the power of $1$ .

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

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

Step 5

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

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

Step 6

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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