Square Root Calculator
Find the principal square root of any number, switch to a cube root or any custom nth root, and see when the result is a perfect power — all in your browser.
Enter a number to find its principal (positive) square root. Switch the root type to compute a cube root or any custom nth root. The tool also flags perfect squares and perfect powers.
√144 = 12
Exact: this is a perfect power — the root is the whole number 12.
The principal square root of x is the non-negative value r with r² = x. An nth root solves rⁿ = x and is computed as x^(1/n). Even roots of negative numbers are not real.
Quick answer
The square root of a number x is the value r with r² = x. The square root of 144 is 12 because 12 × 12 = 144. This calculator returns the principal (positive) root, also handles cube roots and any nth root via x^(1/n), and notes that even roots of negative numbers are not real.
Formula & method
For a square root the tool uses r = √x; for a cube root r = ∛x; and for a custom degree n it computes x^(1/n). It checks whether rounding the result to the nearest whole number reproduces x exactly, flagging perfect squares and perfect powers. Even roots of negative numbers have no real value.
Examples
- Input
- √144
- Result
- 12
- Why
- 12 × 12 = 144, so 144 is a perfect square.
- Input
- √2
- Result
- ≈ 1.41421356
- Why
- 2 is not a perfect square, so its root is an irrational decimal.
- Input
- ∛27 (n = 3)
- Result
- 3
- Why
- 3 × 3 × 3 = 27, so the cube root of 27 is 3.
When to use this tool
- Solving geometry problems involving areas, the Pythagorean theorem, or distances.
- Finding a side length from an area, or a rate from a squared quantity.
- Checking whether a number is a perfect square or perfect power.
Common mistakes
- Forgetting the negative root — every positive number has two square roots (+ and −); this tool reports the principal positive one.
- Expecting a real answer for an even root of a negative number, which is complex (imaginary).
- Confusing the cube root of a negative number (which is real, e.g. ∛−8 = −2) with an even root of a negative number (which is not).
Frequently asked questions
What is a principal square root?
The non-negative root. While both +12 and −12 square to 144, the principal square root of 144 is +12.
Can it do cube roots and other roots?
Yes. Switch the root type to cube root, or choose a custom nth root, and the tool computes x^(1/n).
Why is √2 a long decimal?
Because 2 is not a perfect square, its square root is irrational — a non-terminating, non-repeating decimal, shown here to several places.
What about the square root of a negative number?
It has no real value; the roots are imaginary (for example √−4 = 2i). Odd roots of negatives, however, are real.
How does it detect a perfect square?
It rounds the computed root and checks whether raising that whole number to the power reproduces the original number exactly.
Is anything uploaded?
No. The calculation runs entirely in your browser.
Sources & references
External references open in a new tab. We are independent and not affiliated with these organizations.
- ✓ Free to use
- ✓ No sign-up required
- ✓ Runs entirely in your browser — nothing is uploaded.
- ✓ Formula and method shown above
Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.
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