You can find square roots of perfect squares like $\sqrt{49} = 7$. Most numbers are not perfect squares, but many have a perfect square hiding inside. Pulling it out gives the simplest exact form: $\sqrt{72} = 6\sqrt{2}$.
Why it works
Why may we split $\sqrt{36 \cdot 2}$ into $\sqrt{36} \cdot \sqrt{2}$? Square the product $\sqrt{36} \cdot \sqrt{2}$: you get $\left(\sqrt{36}\right)^2 \cdot \left(\sqrt{2}\right)^2 = 36 \cdot 2 = 72$. It is positive and its square is $72$, so it is $\sqrt{72}$. In general $\sqrt{ab} = \sqrt{a}\sqrt{b}$ for $a, b \ge 0$. Then $\sqrt{36} = 6$ comes out as a whole number, and the $\sqrt{2}$ stays inside.
Perfect squares to look for: $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$, $121$, $144$.
Simplifying is not just tidiness. It reveals when two roots are really the same kind: $\sqrt{8}$ and $\sqrt{18}$ look unrelated, but they are $2\sqrt{2}$ and $3\sqrt{2}$, so their sum is $5\sqrt{2}$. You will use this when adding radicals and when solving quadratic equations.
Key idea
Write the number under the root as (largest perfect square) $\times$ (what is left), and take the square root of the perfect square. A root is simplified when no perfect square except $1$ divides the number inside.
Worked examples
Worked example: The largest square
Simplify $\sqrt{72}$.
- Perfect squares that divide $72$: $4$, $9$, $36$. The largest is $36$, and $72 = 36 \cdot 2$.
- $\sqrt{72} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$.
Answer: $6\sqrt{2}$
Worked example: A smaller square first
Simplify $\sqrt{48}$ if you first notice only the factor $4$.
- $\sqrt{48} = \sqrt{4} \cdot \sqrt{12} = 2\sqrt{12}$.
- $12$ still has the square factor $4$: $2\sqrt{12} = 2 \cdot 2\sqrt{3} = 4\sqrt{3}$.
- Using $16$ from the start gives the same: $\sqrt{16 \cdot 3} = 4\sqrt{3}$.
Answer: $4\sqrt{3}$
Worked example: A number in front
Simplify $3\sqrt{50}$.
- $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.
- $3 \cdot 5\sqrt{2} = 15\sqrt{2}$.
Answer: $15\sqrt{2}$
Common mistake
Splitting a root over a sum: $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 3 + 4 = 7$. The product rule is about multiplication only. Here $\sqrt{9 + 16} = \sqrt{25} = 5$. Check any simplification by squaring your answer: $(6\sqrt{2})^2 = 36 \cdot 2 = 72$, as it should be.