2008 AMC 10B Problem 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.