The one move
Simplifying a square root means pulling out any perfect square hiding inside the number. The rule that lets you do it is:
√(a × b) = √a × √b
That holds for any non-negative a and b. The trick is choosing a to be a perfect square, because then √a comes out as a plain whole number.
Worked: √72
Step 1 — find the largest perfect square that divides 72. The perfect squares are 4, 9, 16, 25, 36, 49, 64. Of those, 36 divides 72 exactly.
Step 2 — split it. 72 = 36 × 2, so √72 = √36 × √2.
Step 3 — take the root you know. √36 = 6.
Answer: 6√2, which is about 8.485281.
Checking it
Square your answer and you should get back to where you started. 6√2 squared is 6² × 2 = 36 × 2 = 72. It checks out.
This is worth doing every time. It catches the most common error — multiplying when you meant to take a root — in about five seconds.
Picking the wrong square is not fatal
Suppose you spot 4 rather than 36. Then √72 = √4 × √18 = 2√18. That is correct, just not finished. Now simplify √18: it is 9 × 2, so √18 = 3√2.
Putting it together: 2 × 3√2 = 6√2. Same answer, one extra step. You cannot go wrong by choosing a small factor, only slowly.
The prime factor method
If spotting perfect squares is not coming naturally, factor the number into primes and look for pairs.
72 = 2 × 2 × 2 × 3 × 3
Group them: (2 × 2) and (3 × 3) are pairs, with a single 2 left over. Each pair contributes one copy of itself outside the root — so 2 and 3 come out, giving 2 × 3 = 6, and the unpaired 2 stays inside.
6√2. This method is slower but it never requires you to spot anything, which makes it the reliable fallback on an exam.
When there is nothing to pull out
Some numbers have no perfect square factor beyond 1. √2, √3, √5, √7 and √11 are already as simple as they get. If you factor a number and find no repeated primes, you are finished before you started.