For all real numbers x > 0, which of the following expressions are equivalent to \left(\fr… — GRE General Test

For all real numbers $x > 0$ , which of the following expressions are equivalent to $\left(\frac{x^{3/2} \cdot \sqrt[3]{x}}{x^{1/6}}\right)^2$ ? Select all such expressions.

$\sqrt[3]{x^{10}}$ Answer
$x^3 \sqrt[3]{x}$ Answer
C
$x^3 \sqrt{x}$
D
$x^5$
$\left(\sqrt[3]{x^5}\right)^2$ Answer

Answer

The expressions equivalent to the given expression are

\sqrt[3]{x^{10}}

x^3 \sqrt[3]{x}

, and

\left(\sqrt[3]{x^5}\right)^2

Simplifying the given expression by converting all radical forms to fractional exponents yields

x^{10/3}

. The three expressions

\sqrt[3]{x^{10}}

x^3 \sqrt[3]{x}

, and

\left(\sqrt[3]{x^5}\right)^2

each rewrite to

x^{10/3}

when evaluated using standard power laws.

Step-by-Step Solution

Convert radical expressions to fractional exponent form inside the parentheses.

\sqrt[3]{x} = x^{1/3}

, so the numerator becomes

x^{3/2} \cdot x^{1/3}

Converting all terms to exponent notation allows applying standard exponent addition and subtraction rules.

Simplify the numerator by adding exponents.

x^{3/2 + 1/3} = x^{9/6 + 2/6} = x^{11/6}

When multiplying exponential terms with the same base, add their exponents using a common denominator.

Divide by the denominator by subtracting exponents.

\frac{x^{11/6}}{x^{1/6}} = x^{11/6 - 1/6} = x^{10/6} = x^{5/3}

When dividing exponential terms with the same base, subtract the denominator's exponent from the numerator's exponent.

Apply the outer exponent of 2.

\left(x^{5/3}\right)^2 = x^{(5/3) \cdot 2} = x^{10/3}

When raising a power to another power, multiply the inner and outer exponents.

Compare

x^{10/3}

to each option.

\sqrt[3]{x^{10}} = x^{10/3}

x^3 \sqrt[3]{x} = x^{3 + 1/3} = x^{10/3}

, and

\left(\sqrt[3]{x^5}\right)^2 = (x^{5/3})^2 = x^{10/3}

are all equivalent.

Matching each candidate expression in fractional exponent form identifies all equivalent choices.

Key Concept

Simplification of algebraic expressions using rules of fractional exponents and radicals

Estimated Time:1m 30s

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