The square root of a number is the value that, multiplied by itself, returns it. Every positive number has two such values, one positive and one negative, and the radical sign refers to the positive one by convention. That convention matters when solving equations: x² = 9 has two solutions, but √9 is 3 alone.
Most roots are irrational — they cannot be written as a fraction and their decimals never settle into a pattern. √2 was the first number proved to have that property, and the proof reportedly caused enough trouble for the Pythagoreans that it became a story. Where the number under the root has a square factor, the surd can be simplified: √72 becomes 6√2, which is exact where a decimal is only an approximation. This page gives both forms.
The formula
n- The number under the root, which must not be negative
√n- The principal, positive square root
perfect square- A number whose root is a whole number
a√b- Simplified surd form, with the largest square factor taken outside
How it works, step by step
- Enter the number you want the root of.
- The gauge shows the principal square root.
- Check below whether the number is a perfect square.
- The simplified surd is the exact form where a decimal would only approximate.
Worked examples
√72 in exact form
72 = 36 × 2, and 36 is a perfect square, so √72 = 6√2 exactly. As a decimal that is 8.485281, which is an approximation no matter how many digits you keep.
Bracketing a root by hand
For √200, note that 14² = 196 and 15² = 225, so the root lies between them. The true value is 14.142136, and the simplified surd is 10√2 since 200 = 100 × 2.
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Frequently asked questions
Does a positive number have two square roots?
Yes, one positive and one negative, since both square to the same value. The radical symbol denotes the positive one, which is why solving x² = 9 needs the explicit ± that √9 does not carry.
What does simplifying a surd achieve?
It separates the exact part from the irrational one. 6√2 is precise, while 8.485281 is rounded. Carrying the surd through a calculation avoids accumulating rounding error.
Why is √2 irrational?
Assume it were a fraction in lowest terms; squaring gives a contradiction, because both numerator and denominator would have to be even and so the fraction was not in lowest terms after all. It is the classic proof by contradiction.
Can I take the square root of a negative number?
Not within the real numbers. Complex numbers define i as the root of −1, so √−9 becomes 3i. This page reports the imaginary form rather than failing.
Square roots and surd forms
| n | Square root | Perfect square? | Simplified surd |
|---|---|---|---|
| 16 | 4.000000 | yes | 4 |
| 20 | 4.472136 | no | 2√5 |
| 50 | 7.071068 | no | 5√2 |
| 72 | 8.485281 | no | 6√2 |
| 100 | 10.000000 | yes | 10 |
| 200 | 14.142136 | no | 10√2 |
| 256 | 16.000000 | yes | 16 |
Where the surd reduces to a whole number, the original was a perfect square.