2003 AMC 10A Problem 9

Question

Simplify

\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}.

\sqrt{x}

\sqrt[3]{x^2}

\sqrt[27]{x^2}

\sqrt[54]{x}

\sqrt[81]{x^{80}}

Rewrite each radical using fractional exponents, working from the inside out

The innermost

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

and each cube root divides the exponent by

3

Working outward,

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

and its cube root is

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

Then

x \cdot x^{\frac{1}{2}} = x^{\frac{3}{2}},

whose cube root is again

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

Repeating once more,

x \cdot x^{\frac{1}{2}} = x^{\frac{3}{2}},

whose cube root is

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

Thus, the correct answer is A.

Problem 9 in other years