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

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

Step 1

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

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

Step 2

Combine $x^{\frac{1}{2}}$ and $\frac{1}{y^{\frac{1}{4}}}$ .

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

Step 3

Combine $\frac{x^{\frac{1}{2}}}{y^{\frac{1}{4}}}$ and $z$ .

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

Step 4

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{y^{\frac{1}{4}}}{x^{\frac{1}{2}} z}$ .

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

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

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

Step 6

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Simplify the denominator.

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

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

$\frac{y^{\frac{1}{2}}}{x^{\frac{1}{2} \cdot 2} z^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{y^{\frac{1}{2}}}{x^{\frac{1}{2} \cdot 2} z^{2}}$

Rewrite the expression.

$\frac{y^{\frac{1}{2}}}{x^{1} z^{2}}$

Simplify.

$\frac{y^{\frac{1}{2}}}{x z^{2}}$