Use the product rule to simplify sqrt(x^3). Assume the variables represent nonnegative real numbers.

Step-by-step solution

1. 1

   Decide what "simplified" means for a radical. A square root is simplified when no perfect-square factor is left under the radical sign. The exponent $3$ is odd, so it cannot come out whole — but the largest even part of it can. That is the entire strategy: split off the biggest perfect square hiding inside $x^{3}$.

2. 2

   Rewrite the radicand as a perfect square times a leftover. Since $3=2+1$,

   $x^{3}=x^{2}\cdot x\qquad\Longrightarrow\qquad \sqrt{x^{3}}=\sqrt{x^{2}\cdot x}$

3. 3

   Apply the product rule for radicals. For nonnegative $a$ and $b$,

   $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$

   so

   $\sqrt{x^{2}\cdot x}=\sqrt{x^{2}}\cdot\sqrt{x}$

   The nonnegativity assumption in the problem statement is exactly what licenses this split.

4. 4

   Simplify the perfect-square factor. In general $\sqrt{x^{2}}=|x|$, but the problem states $x\geq 0$, so the absolute value is unnecessary:

   $\sqrt{x^{2}}=x$

   Hence

   $\sqrt{x^{3}}=x\sqrt{x}$

5. 5

   Confirm the result is fully simplified and check a value. The remaining radicand is $x$ to the first power, which has no perfect-square factor, so the job is done. Numerically, at $x=9$ the original is $\sqrt{729}=27$ and the simplified form is $9\sqrt{9}=9\cdot 3=27$; at $x=2$ both give $\sqrt{8}=2\sqrt{2}\approx 2.8284$.