Simplifying radicals
Simplify a square root to simplest form, e.g. √72 = 6√2, with steps.
How to simplify a radical
We simplify a radical by factoring out the largest perfect square that divides the number under the root, and "pulling it out" of the radical. We use the rule √(a·b) = √a · √b.
√72 = √(36 · 2) = √36 · √2 = 6√2
The number 36 is the largest square dividing 72 (36 = 6²). Its square root is 6, which we put in front of the radical, and the remainder, 2, stays underneath. The simplified form 6√2 is exact – unlike the decimal approximation 8.485…, it loses no information.
What it's good for
The simplified form is used in math and physics when working with exact values (not rounded). It shows up when solving quadratic equations, in trigonometry (√2/2, √3/2), and anywhere a "clean" result without an infinite decimal expansion is needed.
Tip: if the number is itself a perfect square (e.g. 49), the root comes out as a whole number and there is nothing to simplify.