For all positive real numbers x and y, which of the following expressions are equivalent t… — GRE General Test

For all positive real numbers $x$ and $y$ , which of the following expressions are equivalent to $\left(\frac{x^{-2} y^{3/2}}{\sqrt{x^4 y^{-1}}}\right)^{-1/2}$ ? Select all that apply.

$x^2 y^{-1}$ Answer
$\sqrt{\frac{x^4}{y^2}}$ Answer
C
$\frac{y}{x^2}$
$\frac{x^3 y}{\sqrt{x^2 y^4}}$ Answer
E
$\frac{x^2}{\sqrt{y}}$

Answer

The expressions equivalent to the given expression are

x^2 y^{-1}

\sqrt{\frac{x^4}{y^2}}

, and

\frac{x^3 y}{\sqrt{x^2 y^4}}

Simplifying the original expression step-by-step yields

\frac{x^2}{y}

, which is equal to

x^2 y^{-1}

. Taking the square root of

\frac{x^4}{y^2}

gives

\frac{x^2}{y}

, and simplifying

\frac{x^3 y}{\sqrt{x^2 y^4}} = \frac{x^3 y}{x y^2}

also gives

\frac{x^2}{y}

. Thus, these three choices are equivalent to the target expression.

Step-by-Step Solution

Simplify the denominator inside the parentheses

\sqrt{x^4 y^{-1}} = (x^4 y^{-1})^{1/2} = x^2 y^{-1/2}

Apply the power of a product rule and principal square root properties for positive variables.

Simplify the fraction inside the parentheses

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

Subtract exponents of like bases when dividing.

Apply the outer exponent

-1/2

(x^{-4} y^2)^{-1/2} = x^{(-4)(-1/2)} y^{(2)(-1/2)} = x^2 y^{-1} = \frac{x^2}{y}

Multiply exponents when raising a power to a power.

Evaluate each option against

\frac{x^2}{y}

x^2 y^{-1}

\sqrt{\frac{x^4}{y^2}}

, and

\frac{x^3 y}{\sqrt{x^2 y^4}}

all simplify to

\frac{x^2}{y}

Check algebraic equivalence for each provided choice.

Key Concept

Algebraic Exponents and Radicals

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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