which expression is equivalent to $left(x^{\frac{1}{2}}y^{-\frac{1}{4}}z\right)^{-2}$?\n$\frac{x^{\frac{1}{2}…

Explanation:

Step1: Distribute the outer exponent to each factor inside the parentheses.

Apply the power of a product rule $(abc)^n = a^n b^n c^n$. Assuming the expression is $(x^{\frac{1}{2}} y^{-\frac{1}{4}} z)^{-2}$ based on the options provided. $$ (x^{\frac{1}{2}} y^{-\frac{1}{4}} z)^{-2} = (x^{\frac{1}{2}})^{-2} (y^{-\frac{1}{4}})^{-2} (z^1)^{-2} $$

Step2: Multiply the exponents for each factor.

Apply the power of a power rule $(a^m)^n = a^{mn}$. $$ (x^{\frac{1}{2}})^{-2} = x^{\frac{1}{2} \times (-2)} = x^{-1} $$ $$ (y^{-\frac{1}{4}})^{-2} = y^{-\frac{1}{4} \times (-2)} = y^{\frac{2}{4}} = y^{\frac{1}{2}} $$ $$ (z^1)^{-2} = z^{1 \times (-2)} = z^{-2} $$ The expression simplifies to $x^{-1} y^{\frac{1}{2}} z^{-2}$.

Step3: Rewrite the expression using positive exponents.

Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$. $$ x^{-1} y^{\frac{1}{2}} z^{-2} = \frac{1}{x} \cdot y^{\frac{1}{2}} \cdot \frac{1}{z^2} $$

Step4: Combine the terms into a single fraction.

Multiply the fractions. $$ \frac{1 \cdot y^{\frac{1}{2}} \cdot 1}{x \cdot 1 \cdot z^2} = \frac{y^{\frac{1}{2}}}{xz^2} $$

Answer:

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