2008 AMC 10B Problem 3

Problem 3 of 25EasierAlgebra

Assume that

x

is a positive real number. Which is equivalent to

\sqrt[3]{x\sqrt{x}}\,?

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Solution

Since

\sqrt{x}=x^{\frac{1}{2}},

we have

x\sqrt{x}=x^{1}\cdot x^{\frac{1}{2}}=x^{\frac{3}{2}}.

Taking the cube root multiplies the exponent by

\tfrac13,

giving

\left(x^{\frac{3}{2}}\right)^{\frac{1}{3}}=x^{\frac{1}{2}}.

Thus, the correct answer is D.

Concepts: exponent · radical

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