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

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

$\frac{a^2}{b^2}$ Answer
$\left( \frac{b}{a} \right)^{-2}$ Answer
$\sqrt{\frac{a^4}{b^4}}$ Answer
D
$\frac{a^{-2}}{b^{-2}}$
E
$\frac{b^2}{a^2}$

Answer

The expressions equivalent to the given quantity are

\frac{a^2}{b^2}

\left( \frac{b}{a} \right)^{-2}

, and

\sqrt{\frac{a^4}{b^4}}

Simplifying the original expression step-by-step yields

\frac{a^2}{b^2}

. The expressions

\frac{a^2}{b^2}

\left( \frac{b}{a} \right)^{-2} = \left( \frac{a}{b} \right)^2 = \frac{a^2}{b^2}

, and

\sqrt{\frac{a^4}{b^4}} = \frac{a^2}{b^2}

are all identical to the simplified form.

Step-by-Step Solution

Simplify the radical in the denominator of the inner expression.

Since

a > 0

and

b > 0

\sqrt{a^4 b^{-2}} = \sqrt{a^4} \cdot \sqrt{b^{-2}} = a^2 b^{-1}

Applying the square root to each variable power individually.

Simplify the expression inside the outer parenthesis.

\frac{a^{-2} b^3}{a^2 b^{-1}} = a^{-2 - 2} b^{3 - (-1)} = a^{-4} b^4.

Subtracting exponents of like bases according to the quotient rule of exponents.

Apply the outer exponent of

-\frac{1}{2}

(a^{-4} b^4)^{-1/2} = (a^{-4})^{-1/2} (b^4)^{-1/2} = a^2 b^{-2} = \frac{a^2}{b^2}.

Multiplying internal exponents by

-\frac{1}{2}

according to the power rule of exponents.

Evaluate each choice against the simplified form

\frac{a^2}{b^2}

The expressions

\frac{a^2}{b^2}

\left(\frac{b}{a}\right)^{-2}

, and

\sqrt{\frac{a^4}{b^4}}

are all algebraically equivalent to

\frac{a^2}{b^2}

Testing algebraic equivalence using standard exponent and radical laws.

Key Concept

Properties of exponents and radicals, including power of a quotient, quotient rule, negative exponents, and square roots of powers.

Estimated Time:1m 30s

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