We are asked to simplify the expression $\sqrt{80x} - \sqrt{32x} + \sqrt{8x}$, assuming $x$ is a nonnegative real number.

1. Problem Description

We are asked to simplify the expression $\sqrt{80x} - \sqrt{32x} + \sqrt{8x}$, assuming $x$ is a nonnegative real number.

2. Solution Steps

First, we simplify each term:

$\sqrt{80x} = \sqrt{16 \cdot 5x} = \sqrt{16} \cdot \sqrt{5x} = 4\sqrt{5x}$

$\sqrt{32x} = \sqrt{16 \cdot 2x} = \sqrt{16} \cdot \sqrt{2x} = 4\sqrt{2x}$

$\sqrt{8x} = \sqrt{4 \cdot 2x} = \sqrt{4} \cdot \sqrt{2x} = 2\sqrt{2x}$

Now, we substitute these simplified expressions back into the original expression:

$4\sqrt{5x} - 4\sqrt{2x} + 2\sqrt{2x}$

Combine the terms with the same radical:

$4\sqrt{5x} + (-4+2)\sqrt{2x}$

$4\sqrt{5x} - 2\sqrt{2x}$

The terms $4\sqrt{5x}$ and $-2\sqrt{2x}$ cannot be combined further because the terms inside the square root are different.

3. Final Answer

$4\sqrt{5x} - 2\sqrt{2x}$