2008 AMC 10B Problem 3

Roots are easiest to handle as exponents, so rewrite everything that way. Work from the inside out.

1. A square root is the power $\tfrac12$: $\sqrt{x} = x^{1/2}$.
2. Multiplying powers of the same base adds the exponents:
$$x \cdot \sqrt{x} = x^{1} \cdot x^{1/2} = x^{1 + 1/2} = x^{3/2}.$$
3. A cube root is the power $\tfrac13$, and a power of a power multiplies the exponents:
$$\sqrt[3]{x^{3/2}} = x^{\frac32 \cdot \frac13} = x^{1/2}.$$

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

Once every root is written as a power of $x$, only two rules are needed: multiplying adds exponents, and taking a root divides the exponent. Handling the inside first and the outside last keeps the steps in the right order. Whenever roots are stacked inside roots, switch to exponents.

Try a number that makes every root come out whole. With $x = 64$: $\sqrt{64} = 8$, so $x\sqrt{x} = 64 \cdot 8 = 512$, and $\sqrt[3]{512} = 8$ because $8 \cdot 8 \cdot 8 = 512$. Now check the choices with $x = 64$: $64^{1/2} = 8$, while $64^{1/6} = 2$, $64^{1/4}$ and $64^{3/8}$ are not whole numbers, and $64^{1} = 64$. Only (D) gives $8$.

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