Simplifying Square Roots
Simplify square root expressions by factoring out perfect squares. Distinguish between rational and irrational square roots and simplify expressions like √72 = 6√2.
A lesson of Algebra 1, in Radical Expressions & Equations, about 30 minutes.
By the end of this lesson, you will be able to pull perfect squares out of a radical, simplify by prime factorization when the number is large, tell a rational root from an irrational one, and say why the exact form beats a decimal.
The Product Rule Is the Whole Method
$$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$
A root splits across a product. That single rule is everything in this lesson, because it lets you separate the part of a number that has a whole square root from the part that does not.
$$\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$$
The $4$ came out as a $2$. The $3$ stayed inside, because nothing whole squares to $3$.
Look for the LARGEST perfect square that divides the number. $\sqrt{72}$ can be split as $\sqrt{4 \cdot 18} = 2\sqrt{18}$, which is correct but unfinished: $18$ still contains a $9$. Splitting off $36$ instead gives $6\sqrt{2}$ in one step.
The rule does not work across addition.
$$\sqrt{9 + 16} = \sqrt{25} = 5 \qquad \text{but} \qquad \sqrt{9} + \sqrt{16} = 7$$
Products split; sums do not. That is the most expensive mistake in the topic and it is worth checking with those exact numbers whenever you doubt it.
Perfect Squares Worth Knowing
$$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225$$
Recognizing these on sight is what makes simplifying fast. A few results that come up constantly:
| Radical | Split as | Simplified |
|---|---|---|
| $\sqrt{8}$ | $4 \cdot 2$ | $2\sqrt{2}$ |
| $\sqrt{18}$ | $9 \cdot 2$ | $3\sqrt{2}$ |
| $\sqrt{20}$ | $4 \cdot 5$ | $2\sqrt{5}$ |
| $\sqrt{32}$ | $16 \cdot 2$ | $4\sqrt{2}$ |
| $\sqrt{50}$ | $25 \cdot 2$ | $5\sqrt{2}$ |
| $\sqrt{75}$ | $25 \cdot 3$ | $5\sqrt{3}$ |
| $\sqrt{98}$ | $49 \cdot 2$ | $7\sqrt{2}$ |
A radical is fully simplified when what remains inside has no perfect-square factor left. $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$ and $\sqrt{10}$ are all as simple as they get.
Sort It: Fully Simplified or Not
The fourth and sixth are the ones to notice. A number already outside the root does not mean the job is done; what matters is only what is left inside.