How to use it
- Type the number, for example 72. Decimals work with a dot or a comma.
- Choose square root, cube root or Other and type the index n.
- Read the decimal root, the simplified radical and the steps.
How it works
The decimal value is the n-th root of the number to 10 decimal places. Whether the root is a whole number is checked with exact integer arithmetic, not by rounding, so √144 is reported as exactly 12 and √145 as not exact.
For the simplified form the number is split into prime factors. Each prime’s exponent is divided by the index n: the whole part of the division goes outside the root, the remainder stays inside.
ⁿ√(a^n × b) = a × ⁿ√b
Worked example
√72:
- 72 = 2³ × 3².
- Grouped in pairs: 2² × 2 × 3²; the 2 and the 3 of the pairs come out of the root as 2 × 3 = 6.
- √72 = 6√2 ≈ 8.4852813742.
- Check: 6² × 2 = 36 × 2 = 72.
The cube root of 54 works the same way with groups of three: 54 = 3³ × 2, so ∛54 = 3∛2.
Good to know
- Every positive number has two square roots, for example 3 and −3 for 9. The calculator shows the positive one (the principal root).
- Perfect squares up to 400: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.