How to Calculate and Simplify a Square Root

Formula

√x = a × √b, where a² × b = x

• x: the number you take the root of; it must be 0 or more
• a: the largest whole number whose square divides x; it goes outside the root
• b: x divided by a²; it stays under the root and has no square factor left

When b = 1, x is a perfect square and the square root is simply a.

Step-by-step example

Simplify √180.

1. Find the largest perfect square that divides 180. Try perfect squares such as 4, 9, 16, 25 and 36: the largest that divides 180 is 36, which is 6². So 180 = 36 × 5.
2. Take the root of the perfect square outside: √180 = √(6² × 5) = 6 × √5.
3. The 5 left under the root has no square factor, so you are done.
4. Read the decimal value: the square root calculator gives 13.416407864998739 for √180, about 13.416.

Common square roots

Common mistakes

• Splitting a root over a sum. √9 + √16 is not √25: √9 = 3 and √16 = 4 add up to 7, while √25 is 5.
• Stopping too early. √72 = 2 × √18 is true but not the simplest form, because 18 still contains 9 = 3². The simplest form is 6 × √2.
• Looking for the root of a negative number. No real number has a negative square, so negative numbers have no real square root.
• Trying to simplify a decimal. The calculator gives the root of a decimal but does not write it as a√b: √0.25 = 0.5.