Simplify ((y^(1/4))/(y^(3/2)))^2

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

Step 1

Move $y^{\frac{1}{4}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 2

Multiply $y^{\frac{3}{2}}$ by $y^{- \frac{1}{4}}$ by adding the exponents.

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

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

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

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

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $2$ .

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

Subtract $1$ from $6$ .

${(\frac{1}{y^{\frac{5}{4}}})}^{2}$

Step 3

Apply the product rule to $\frac{1}{y^{\frac{5}{4}}}$ .

$\frac{1^{2}}{{(y^{\frac{5}{4}})}^{2}}$

Step 4

One to any power is one.

$\frac{1}{{(y^{\frac{5}{4}})}^{2}}$

Step 5

Multiply the exponents in ${(y^{\frac{5}{4}})}^{2}$ .

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

$\frac{1}{y^{\frac{5}{4} \cdot 2}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{1}{y^{\frac{5}{2 (2)} \cdot 2}}$

Cancel the common factor.

$\frac{1}{y^{\frac{5}{2 \cdot 2} \cdot 2}}$

Rewrite the expression.

$\frac{1}{y^{\frac{5}{2}}}$