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

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

Step 1

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

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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

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

Multiply the exponents in ${(y^{\frac{3}{2}})}^{4}$ .

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $2$ .

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

Multiply the exponents in ${(x^{\frac{1}{2}})}^{4}$ .

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{y^{6}}{x^{2}}$