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

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

Step 1

Move $y^{\frac{3}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 2

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

Move $y^{- \frac{3}{2}}$ .

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

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

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

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

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

Combine $3$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $2$ .

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

Add $- 3$ and $6$ .

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

Step 3

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

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

Step 4

Multiply the exponents in ${(x^{\frac{1}{2}})}^{3}$ .

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

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

Combine $\frac{1}{2}$ and $3$ .

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

Step 5

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

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

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

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

Combine $\frac{3}{2}$ and $3$ .

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

Multiply $3$ by $3$ .

$x^{\frac{3}{2}} y^{\frac{9}{2}}$