If x > 1, which of the following expressions are equivalent to \frac{\sqrt{x^5 \cdot x^3}}… — GRE General Test

If $x > 1$ , which of the following expressions are equivalent to $\frac{\sqrt{x^5 \cdot x^3}}{(x^{-2})^{-1}}$ ? Select all such expressions.

The expressions equivalent to the target expression are

(x^{\frac{2}{3}})^3

and

\frac{x^{-1}}{x^{-3}}

The original expression simplifies step by step to

x^2

. The expression

(x^{2/3})^3

equals

x^{(2/3)\cdot 3} = x^2

, and the expression

\frac{x^{-1}}{x^{-3}}

equals

x^{-1 - (-3)} = x^2

. Both are equal to

x^2

Simplify the numerator of the given expression

\sqrt{x^5 \cdot x^3} = \sqrt{x^{5+3}} = \sqrt{x^8} = x^{8/2} = x^4

Combine bases using product rule for exponents, then convert radical to fractional exponent.

Simplify the denominator of the given expression

(x^{-2})^{-1} = x^{(-2) \cdot (-1)} = x^2

Multiply exponents when raising a power to a power.

Divide the simplified numerator by the simplified denominator

\frac{x^4}{x^2} = x^{4-2} = x^2

Subtract denominator exponent from numerator exponent using the quotient rule.

Evaluate each choice against

x^2

Only

(x^{2/3})^3 = x^2

and

\frac{x^{-1}}{x^{-3}} = x^2

are equal to

x^2

Apply basic exponent rules to each proposed expression.

Algebraic Exponents and Radicals Rules

Question