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

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

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

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

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

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

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

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

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

Multiply $3$ by $2$ .

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

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

${(y^{\frac{4 - 3}{6}})}^{2}$

Subtract $3$ from $4$ .

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

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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