Simplify a square root: hunting for perfect squares

The principle

$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

So if $a$ is a perfect square (4, 9, 16, 25, 36, 49, 64, 81, 100, …), you can pull it out:

$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b} = (\text{integer}) \times \sqrt{b}$

The method

Step 1: factor the number under the root using the largest perfect square that divides it.

Step 2: pull the perfect square out of the root (it becomes an integer).

Example 1: $\sqrt{72}$

Largest square dividing 72: 36. $72 = 36 \times 2$.

$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

Example 2: $\sqrt{45}$

$45 = 9 \times 5$.

$\sqrt{45} = 3\sqrt{5}$

Example 3: $\sqrt{200}$

$200 = 100 \times 2$.

$\sqrt{200} = 10\sqrt{2}$

Handy table

Perfect square	Root
4	2
9	3
16	4
25	5
36	6
49	7
64	8
81	9
100	10
121	11
144	12

Fallback trick

Prime factorize. Every pair of the same factor = a square to pull out.

$\sqrt{300} = \sqrt{2 \times 2 \times 3 \times 5 \times 5} = \sqrt{(2 \times 5)^2 \times 3} = 10\sqrt{3}$

?À toi de jouer

Simplify √98.

A.7√2B.√98 (cannot be simplified)C.14√2D.√49 + √2

Valider