Simplify Square Roots

How it works

Find the largest perfect square that divides the number, and take its root outside: √180 = √36 × √5 = 6√5. A root is fully simplified when nothing under it has a square factor left.

Example

Simplify √180.

√180 = 6√5

Decimal

13.416408

Steps

1. 1
   Largest perfect square that divides 180: 36, because 180 = 36 × 5.
2. 2
   √180 = √36 × √5.
3. 3
   √36 = 6, so √180 = 6√5.

More examples

Simplify √72.

√72 = 6√2

Decimal

8.485281

Steps

1. 1
   Largest perfect square that divides 72: 36, because 72 = 36 × 2.
2. 2
   √72 = √36 × √2.
3. 3
   √36 = 6, so √72 = 6√2.

Simplify √252.

√252 = 6√7

Decimal

15.874508

Steps

1. 1
   Largest perfect square that divides 252: 36, because 252 = 36 × 7.
2. 2
   √252 = √36 × √7.
3. 3
   √36 = 6, so √252 = 6√7.

Common mistake

Roots split over multiplication, not addition. √(9 + 16) is 5, while √9 + √16 is 7.

Questions

How do I know a root is fully simplified?

Nothing under the root has a square factor (4, 9, 16, 25, …) left.

Which perfect squares should I look for?

4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. Try the largest one that divides the number first.

How do I multiply roots?

Multiply under one root, then simplify: √6 × √12 = √72 = 6√2.

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