For all x > 1, which of the following expressions is equivalent to \frac{\sqrt{x^3 \cdot \… — GRE General Test

For all $x > 1$ , which of the following expressions is equivalent to $\frac{\sqrt{x^3 \cdot \sqrt[3]{x^2}}}{(x^{-1/3})^2}$ ?

x^{5/2}

Converting all radical expressions into rational exponent form simplifies the numerator to

(x^{11/3})^{1/2} = x^{11/6}

and the denominator to

x^{-2/3}

. Applying the quotient rule

x^{11/6} / x^{-2/3} = x^{11/6 - (-2/3)}

yields

x^{15/6} = x^{5/2}

Convert radicals in the numerator to fractional exponents and combine terms inside the square root

x^3 \cdot \sqrt[3]{x^2} = x^3 \cdot x^{2/3} = x^{3 + 2/3} = x^{11/3}

When multiplying exponential expressions with the same base, add the exponents.

Apply the outer square root to the simplified expression in the numerator

\sqrt{x^{11/3}} = (x^{11/3})^{1/2} = x^{11/6}

Taking the square root is equivalent to raising an expression to the power of

1/2

Simplify the denominator using the power of a power rule

(x^{-1/3})^2 = x^{(-1/3) \cdot 2} = x^{-2/3}

When raising a power to another power, multiply the exponents.

Divide the simplified numerator by the simplified denominator

\frac{x^{11/6}}{x^{-2/3}} = x^{11/6 - (-2/3)} = x^{11/6 + 4/6} = x^{15/6} = x^{5/2}

When dividing exponential expressions with the same base, subtract the denominator exponent from the numerator exponent.

Simplifying nested algebraic radicals using rational exponent laws

Estimated Time:1m 30s

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