1. **State the problem:** We are given the function $F(x) = x \sqrt{x} - 1$. We want to understand and simplify this function. 2. **Rewrite the function:** Recall that $\sqrt{x} = x^{1/2}$. Therefore, $$F(x) = x \cdot x^{1/2} - 1 = x^{1 + 1/2} - 1 = x^{3/2} - 1.$$ 3. **Simplify interpretation:** The function can be written as $$F(x) = x^{3/2} - 1,$$ which is a power function shifted down by 1. 4. **Explain the domain:** Since $\sqrt{x}$ requires $x \geq 0$, the domain of $F(x)$ is $x \geq 0$. 5. **Summary:** The function $F(x)$ simplifies to $x^{3/2} - 1$ for $x \geq 0$.