Square Root Calculator

How to use the square root calculator

1. Type a number. Whole numbers, decimals and negative numbers all work.
2. Read the square root. The answer appears as you type, shown to 6 decimal places by default.
3. Check the simplest radical form. For whole numbers that are not perfect squares, the calculator also shows the root as a simplified radical such as 5√2.
4. See whether it is a perfect square. The calculator says Yes or No and, if No, shows the two perfect squares the number sits between.
5. Change the precision (optional). Open Advanced options to show anywhere from 0 to 12 decimal places.

For the square root of a fraction, take the square root of the numerator and denominator separately, or divide the numerator by the denominator and enter the decimal.

What is a square root?

The square root of a number is the value that gives the number when it is multiplied by itself. It is written with the radical sign √.

√x = y means y × y = x

For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots, a positive one and a negative one (5 and −5), and the radical sign means the positive one, called the principal square root.

Perfect squares

A perfect square is a whole number whose square root is also a whole number:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

So √144 = 12 and √225 = 15. Any other whole number has a square root that is irrational: a decimal that never ends and never repeats.

How to simplify a square root

To write a square root in its simplest radical form, pull out the largest perfect square that divides the number.

Example: simplify √50

1. Find the largest perfect square factor of 50. It is 25, because 50 = 25 × 2.
2. Split the root: √50 = √25 × √2
3. Take the square root of the perfect square: √25 = 5
4. The simplest form is 5√2, which is about 7.0711.

If no perfect square factor exists, the number is already in simplest form. For example, √15 stays √15 because 15 = 3 × 5 has no square factor.

How to find a square root without a calculator

The Babylonian method converges very quickly. To find √20:

1. Make a guess, for example 4.
2. Divide the number by the guess: 20 ÷ 4 = 5.
3. Average the guess and the result: (4 + 5) ÷ 2 = 4.5.
4. Repeat with the new guess: 20 ÷ 4.5 = 4.444, so the average is 4.4722.
5. One more round gives 4.472136, which is √20 to six decimals.

Common square roots

• √2 ≈ 1.414214
• √3 ≈ 1.732051
• √5 ≈ 2.236068
• √10 ≈ 3.162278
• √20 = 2√5 ≈ 4.472136
• √50 = 5√2 ≈ 7.071068
• √100 = 10

Square roots of negative numbers

No real number multiplied by itself gives a negative result, so the square root of a negative number is imaginary. It is written with i, where i = √−1. For example, √−16 = 4i and √−50 = 5i√2.