2008 AMC 10B Problem 3

Question

Assume that

x

is a positive real number. Which is equivalent to

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

x^{\frac{1}{6}}

x^{\frac{1}{4}}

x^{\frac{3}{8}}

x^{\frac{1}{2}}

x

Write

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

Then

x\sqrt{x}=x^{\frac{3}{2}},

and the cube root divides the exponent by

3

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.

Problem 3 in other years