1. **Stating the problem:** Simplify and understand the expression $\sqrt{x}$. 2. **Formula and rules:** The square root function $\sqrt{x}$ is defined as the non-negative number which, when squared, gives $x$. That is, if $y = \sqrt{x}$, then $y^2 = x$ and $y \geq 0$. 3. **Important notes:** - The domain of $\sqrt{x}$ is $x \geq 0$ because the square root of a negative number is not a real number. - $\sqrt{x^2} = |x|$ because squaring removes the sign. 4. **Example simplification:** If you have $\sqrt{16}$, then $\sqrt{16} = 4$ because $4^2 = 16$. 5. **Intermediate work:** For example, simplify $\sqrt{25x^2}$: $$\sqrt{25x^2} = \sqrt{25} \times \sqrt{x^2} = 5 \times |x| = 5|x|$$ 6. **Explanation:** The square root function returns the principal (non-negative) root. When dealing with variables inside the root, remember to consider absolute values when simplifying squares. **Final answer:** $\sqrt{x}$ is the principal square root of $x$, defined for $x \geq 0$.