The problem asks to simplify the expression $\sqrt{64x^2}$, assuming that the variable $x$ represents any positive or negative real number.

1. Problem Description

The problem asks to simplify the expression $\sqrt{64x^2}$, assuming that the variable $x$ represents any positive or negative real number.

2. Solution Steps

We need to simplify the square root of the given expression.

First, we can rewrite the expression as:

$\sqrt{64x^2} = \sqrt{64} \cdot \sqrt{x^2}$

We know that $\sqrt{64} = 8$.

Since we are told that $x$ can represent any positive or negative real number, $\sqrt{x^2}$ is equal to the absolute value of $x$, i.e., $|x|$.

So, $\sqrt{x^2} = |x|$.

Therefore, the simplified expression is:

$\sqrt{64x^2} = 8|x|$.

3. Final Answer

8|x|