Square Root Calculator
Square roots, cube roots and any nth root, with the exact value when the root is a whole number.
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Your details
Result
- √144
- 12
- Check
- 12^2 = 144
- Both roots
- ±12
An exact whole number
Every positive number has a negative square root too
The square root of 144…
ⁿ√x = x^(1/n)
n = 2, so 1 ÷ n = 0.5
144^0.5
√144 = 12
Formula — written step by step
The square root of 144…
ⁿ√x = x^(1/n)
n = 2, so the exponent is 1 ÷ 2 = 0.5
144^0.5
Check: 12 × 12 = 144
√144 = 12
x is the number under the radical and n the degree of the root. A root asks the reverse of a power: not "what is 12 squared" but "what squares to 144".
A worked example
A cube root of a negative number
- Number
- −125
- Root
- Cube (3rd)
- An odd root of a negative number is real
- Find the root of the magnitude: ³√125 = 5
- Restore the sign, because odd powers keep it
- Check: (−5) × (−5) × (−5) = −125
³√(−125) = −5
Odd roots of negative numbers are perfectly real; even roots are not. There is no real number that squares to −125, because squaring any real number gives a positive result. This is why the calculator accepts the cube root here but refuses the square root.
How to use this calculator
- Enter the number you want the root of.
- Choose the degree — square, cube, or a custom value for anything else.
- Read the result. Where the root is a whole number the calculator says so; otherwise the value is irrational and shown to ten decimal places.
- Use the check row to confirm the answer: raising it back to the same power should return your original number.
Roots as inverse powers
A root undoes a power. If 12² = 144, then √144 = 12. Formally, the nth root of x is x raised to the power of one over n, which is why every root can be written as an exponent and handled by the same rules.
That equivalence is more than notation. It explains why √a × √b = √(ab) — it is just the product rule for exponents with a fractional power — and it is how calculators evaluate roots internally.
Why even roots reject negatives
Squaring any real number gives a positive result: a positive times a positive is positive, and so is a negative times a negative. Nothing real squares to −4, so √(−4) has no real value, and the same holds for every even root.
Odd roots have no such problem. Cubing a negative number keeps it negative, because three negatives multiply to a negative, so ³√(−125) = −5 is a genuine real answer. JavaScript’s built-in power function returns NaN here — it cannot raise a negative base to a fractional exponent — so this calculator handles the sign separately rather than reporting a false error.
Irrational roots
Most roots are irrational: their decimal expansion never terminates and never repeats. √2 is the classic example, and the proof that it cannot be written as a fraction is one of the oldest results in mathematics.
A displayed value is therefore always an approximation, however many digits are shown. For exact work, leave the root in radical form — √2 rather than 1.4142135624 — and only convert to a decimal at the very end, so rounding error has no chance to accumulate.
Perfect squares and cubes
| n | n² | √n | n³ | ³√n |
|---|---|---|---|---|
| 2 | 4 | 1.414214 | 8 | 1.259921 |
| 3 | 9 | 1.732051 | 27 | 1.442250 |
| 4 | 16 | 2 | 64 | 1.587401 |
| 5 | 25 | 2.236068 | 125 | 1.709976 |
| 9 | 81 | 3 | 729 | 2.080084 |
| 12 | 144 | 3.464102 | 1728 | 2.289428 |
| 16 | 256 | 4 | 4096 | 2.519842 |
Reading the table the other way gives the roots: because 12² = 144, √144 = 12. Only a handful of integers have whole-number roots — every other square root is irrational.
Frequently asked questions
Does every positive number have two square roots?
Yes. Both 12 and −12 square to 144. The radical symbol conventionally denotes the positive root, which is why the calculator shows 12 as the answer and lists ±12 separately.
Why can I not take the square root of a negative number?
Because no real number squares to a negative. The answer exists in the complex numbers as an imaginary value — √(−4) = 2i — but it is not a real number and cannot be placed on a number line.
What is the difference between a root and a power?
They are inverse operations expressed the same way. A power multiplies a number by itself repeatedly; a root asks what number, multiplied by itself that many times, gives your number.
Can the root degree be a fraction?
Yes. A root of degree 0.5 is equivalent to squaring, since x^(1/0.5) = x². The calculator accepts any non-zero degree.