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Simplifying Square Roots — Definition, Formula & Examples

Simplifying square roots means rewriting a square root so that no perfect square factor (other than 1) remains under the radical sign. For example, 72\sqrt{72} simplifies to 626\sqrt{2}.

A square root n\sqrt{n} is in simplified form when the radicand nn contains no perfect square divisor greater than 1. To simplify, factor the radicand as a product a2⋅ba^2 \cdot b where a2a^2 is the largest perfect square factor, then apply the property a2⋅b=ab\sqrt{a^2 \cdot b} = a\sqrt{b}.

Key Formula

a2⋅b=ab\sqrt{a^2 \cdot b} = a\sqrt{b}
Where:
  • a2a^2 = The largest perfect square factor of the radicand
  • bb = The remaining factor after dividing by a², with no perfect square factor other than 1
  • aa = The positive square root of the perfect square factor

How It Works

Start by finding the largest perfect square that divides evenly into the number under the radical. Factor the radicand into that perfect square times whatever is left. Then take the square root of the perfect square and move it in front of the radical sign. The remaining factor stays inside. If you cannot spot the largest perfect square right away, you can break the radicand into smaller factors step by step and pull out pairs.

Worked Example

Problem: Simplify 72\sqrt{72}.
Step 1: Find the largest perfect square that divides 72. The perfect squares are 4, 9, 36. The largest one that divides 72 is 36.
72=36×272 = 36 \times 2
Step 2: Rewrite the square root using this factorization.
72=36×2\sqrt{72} = \sqrt{36 \times 2}
Step 3: Apply the product rule for radicals: split the square root and simplify the perfect square part.
36×2=36⋅2=62\sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}
Answer: 72=62\sqrt{72} = 6\sqrt{2}

Another Example

Problem: Simplify 200\sqrt{200}.
Step 1: Identify the largest perfect square factor of 200. Since 200 = 100 × 2 and 100 is a perfect square, use that.
200=100×2200 = 100 \times 2
Step 2: Split the radical and simplify.
200=100×2=100⋅2=102\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}
Answer: 200=102\sqrt{200} = 10\sqrt{2}

Why It Matters

Simplifying square roots is a core skill in Algebra 1 and appears constantly when you solve quadratic equations using the quadratic formula. Geometry problems involving the Pythagorean theorem often produce unsimplified radicals that need to be reduced. Mastering this technique also prepares you for working with radical expressions in Algebra 2 and precalculus.

Common Mistakes

Mistake: Not using the largest perfect square factor, so the answer still needs further simplification (e.g., writing 72=218\sqrt{72} = 2\sqrt{18} instead of 626\sqrt{2}).
Correction: Always look for the largest perfect square factor first. If you miss it, keep simplifying until no perfect square greater than 1 remains under the radical.
Mistake: Trying to simplify by splitting a sum under the radical, like writing 9+16=9+16\sqrt{9 + 16} = \sqrt{9} + \sqrt{16}.
Correction: The product rule ab=a⋅b\sqrt{ab} = \sqrt{a}\cdot\sqrt{b} works for multiplication, not addition. 9+16=25=5\sqrt{9+16} = \sqrt{25} = 5, which is not 3+4=73 + 4 = 7.

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