A square root calculator finds the number that, multiplied by itself, gives your number. Enter any number and choose how many decimal places you want. The calculator returns the square root, the simplified radical form for whole numbers, the two perfect squares the number sits between, and the steps.
It also covers common variations such as the square root of a decimal, the square root of a non-perfect square, and the square root of a negative number, which is written with i.
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How the square root calculator works
What a square root is
The square root of x, written √x, is the number that multiplies by itself to make x.
√x = y means y × y = x
For example, √144 = 12 because 12 × 12 = 144.
Perfect squares
A perfect square is the square of a whole number. Its square root is a whole number.
| Number | Square |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
Simplifying a square root
To simplify √n, find the largest perfect square that divides n, then take its square root outside the radical.
√(a² × b) = a × √b
Finding the largest perfect square factor
Try the perfect squares from the table above, largest first, and see which one divides n evenly. The calculator does this for whole numbers up to one trillion.
Worked examples
Example 1: √72
72 = 36 × 2, and 36 is a perfect square.
√72 = √36 × √2 = 6√2 ≈ 8.485281
The number sits between 8² = 64 and 9² = 81, so its square root is between 8 and 9.
Example 2: √200
200 = 100 × 2, so √200 = 10√2. To four decimal places it is 14.1421, between 14² = 196 and 15² = 225.
Example 3: A perfect square, √144
144 is 12 × 12, so √144 = 12 exactly and there is no radical left.
Example 4: A negative number, √(−9)
No real number multiplies by itself to give −9. The square root is written with i, where i² = −1, so √(−9) = 3i.
Estimating a square root by hand
Start with a guess g, then improve it with the average of g and n ÷ g.
new guess = (g + n ÷ g) ÷ 2
For √72, start with 8. The first improvement is (8 + 72 ÷ 8) ÷ 2 = (8 + 9) ÷ 2 = 8.5. The next is (8.5 + 72 ÷ 8.5) ÷ 2 ≈ 8.4853, already close to the true value of 8.485281.
Frequently asked questions
How do you find the square root of a number?
Find the number that multiplies by itself to give it. For perfect squares, use the table above. Otherwise estimate between two perfect squares and refine, or use the calculator.
What is the square root of 2?
It is about 1.414214. It never ends and never repeats, so it is written as √2 when an exact value is needed.
How do you simplify a square root?
Divide out the largest perfect square. For √72 that is 36, which gives 6√2.
Is a square root always smaller than the number?
Not always. For numbers between 0 and 1 the square root is larger, for example √0.25 = 0.5.
Can you take the square root of a negative number?
Not among the real numbers. The result is an imaginary number written with i, such as √(−9) = 3i.