Simplifying Expressions with Square Roots
Big Idea: Square roots can be simplified, combined, and manipulated using specific rules — just like fractions or algebraic expressions. Mastering these techniques unlocks your ability to work with irrational numbers cleanly and precisely.
1. Review: What Is a Square Root?
The square root of a number $a$ is a value $b$ such that $b^2 = a$. We write $\sqrt{a}$. For example, $\sqrt{25} = 5$ because $5^2 = 25$.
Key Property: The Product Rule for Radicals
$$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \quad \text{(where } a \geq 0, \; b \geq 0\text{)}$$
This rule works in both directions — and that's the foundation for everything below.