How to Simplify Square Roots, Step by Step

A square root is simplified when no perfect square (4, 9, 16, 25, 36, …) is left inside it. Simplifying turns √72 into 6√2, which is the same number in a cleaner form.

Try it with your own problem ↓

The steps

Find the largest perfect square that divides the number under the root.
Write the number as that perfect square times what is left: √(36 × 2).
Split into two roots: √36 × √2.
Take the square root of the perfect square: 6√2.
If a number was already outside the root, multiply it by the one you pulled out.

A worked example

Simplify √72.

The perfect squares up to 72 are 4, 9, 16, 25, 36 and 49. Of these, 4, 9 and 36 divide 72. The largest is 36.
72 = 36 × 2, so √72 = √(36 × 2).
Split: √36 × √2.
√36 = 6, so the answer is 6√2.
Check: 6√2 ≈ 6 × 1.41421 = 8.485, and √72 ≈ 8.485. ✓

√72 = 6√2.

Try it with your own problem

Open the free The Root Cellar with this example already filled in. It shows every step, and you can type over it with your own numbers.

Work it out: √72Opens The Root Cellar with this example

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Common mistakes

Stopping early. √72 = 2√18 is partly done, but 18 still has a 9 in it. Use the largest perfect square.
Adding under the root. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5.
Dropping what's outside. 3√20 = 3 × 2√5 = 6√5.
Thinking √ means both signs. √36 = 6. The equation x² = 36 has two answers, ±6.

Questions people ask

How do I add square roots?

Simplify first, then add the ones with the same root, like terms. √12 + √27 = 2√3 + 3√3 = 5√3.

What is rationalizing the denominator?

Removing a root from the bottom of a fraction by multiplying the top and bottom by it: 5/√3 = 5√3/3.

How do I know if a number is a perfect square?

If you recognize it from the list of squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …), or its prime factorization has every prime an even number of times.

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