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2008 AMC 10B Problem 3

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2008 AMC 10B Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Exponents, Logarithms & Radicals · difficulty 1 of 5 · about 1 min · position P1-10

Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}$ ?

A)
$x^{1/6}$
B)
$x^{1/4}$
C)
$x^{3/8}$
D)
$x^{1/2}$
E)
$x$

Answer

Key insight

Convert to fractional exponents: x times sqrt(x) is x^(3/2), and the cube root divides the exponent by 3.

Solution

Write every radical as a fractional power. Inside the cube root,
$x\sqrt{x} = x^{1} \cdot x^{1/2} = x^{3/2}.$
Taking the cube root multiplies the exponent by $\tfrac{1}{3}$ :
$\sqrt[3]{x^{3/2}} = x^{(3/2)\cdot(1/3)} = x^{1/2}.$

The answer is $\boxed{\textbf{(D)}\ x^{1/2}}$ .

Why this works

Nested radicals become trivial once everything is a power of the same base: multiplication adds exponents, and an $n$ -th root divides the exponent by $n$ . Doing the inside first and the outer root last mirrors the order of operations.

Alternative approach

Plug in $x = 64$ : $\sqrt{64} = 8$ , $64 \cdot 8 = 512$ , $\sqrt[3]{512} = 8$ , and $8 = 64^{1/2}$ . Only choice (D) gives $8$ .

The trap

Adding exponents incorrectly (x times x^(1/2) as x^(1/2)) or multiplying 1/2 by 1/3 to get x^(1/6).

Common mistakes

Adding exponents incorrectly (x times x^(1/2) as x^(1/2)) or multiplying 1/2 by 1/3 to get x^(1/6).
Treating $\sqrt[3]{x\sqrt{x}}$ as $\sqrt[3]{x}\cdot\sqrt{x} = x^{5/6}$ by mixing up which operation is inside which.

Techniques

Set up the equation/formula and compute; no special trick needed