Square roots with exact simplified form — surds kept exact, plus the decimal value.
Square roots are kept exact where possible (for example, √72 = 6√2) and the decimal value is shown alongside.
Exact simplified form
—How to use this tool
Choose square root or cube root mode.
Type any number — for square roots it must be zero or positive.
The exact simplified form and the decimal value appear instantly, with a squared (or cubed) check.
Switch on neighbours in Advanced to see the perfect squares your number sits between.
Formula & worked example
√72: the largest square dividing 72 is 36 = 6², so √72 = 6√2 exactly, and the decimal value is about 8.4853. Check: 8.4853² ≈ 72. √49 = 7 exactly, because 49 is a perfect square. ∛27 = 3 exactly, because 27 is a perfect cube.
Limitations — read this before you rely on it
Square roots of negative numbers are not real numbers and are refused with an explanation (cube roots of negatives are fine). Non-perfect roots are irrational: the decimal shown is an approximation rounded to 4 decimal places, while the surd form is exact. Very large integers may lose precision in the decimal display.
Frequently asked questions
The square root of a number is the value that, multiplied by itself, gives that number. √49 = 7 because 7 × 7 = 49.
Yes, where possible: √72 is kept as 6√2 rather than only 8.4853, because the exact surd form is precise while the decimal is an approximation.
Only perfect squares have whole-number square roots: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. The calculator tells you when your number is one.
No — in real numbers, a negative number has no square root. Switch to cube root mode: negative numbers do have real cube roots, for example ∛−8 = −2.
Yes, switch to cube root mode. ∛27 = 3 because 3 × 3 × 3 = 27.