√ Square Root Calculator
Find the square root, cube root or any nth root of a number, simplify radicals like √72 = 6√2, handle negatives with i, and see the Babylonian method.
It lies between the perfect squares 8² = 64 and 9² = 81.
The equation x² = 72 has two solutions, x = ±8.48528137423857; the square root symbol means the positive one.
Babylonian (Newton) method
Start with a guess g, then repeat g = (g + 72 ÷ g) ÷ 2. Each step roughly doubles the number of correct digits.
- 36
- 19
- 11.3947368421053
- 8.85672176978242
- 8.49307025882974
- 8.48528494578109
- 8.48528137423932
What Square Root Calculator Does
This square root calculator gives the root of any number as a decimal to 15 significant figures and, for whole numbers, in simplest radical form — √72 = 6√2 — the form most algebra classes ask for. It also shows which two perfect squares the number sits between and the first steps of the Babylonian method, so you can see how a root is found by hand.
Switch the root to cube, 4th, 5th, 6th or 10th to find other roots with the same simplification: ∛54 = 3∛2. A negative number has a real cube root (∛−8 = −2) but no real square root; for even roots of negatives the answer is given as an imaginary number using i = √−1.
How to Use Square Root Calculator
- Enter a number (whole, decimal or negative)
- Choose square root, cube root or another root
- Read the decimal result and the simplified radical form
- See the nearest perfect squares and the Babylonian method steps
Formula Used by Square Root Calculator
Simplifying a radical
√(a² × b) = a√b, where a² is the largest perfect square dividing the number
Worked example
√72
- 72 = 36 × 2, and 36 = 6² is the largest perfect square factor
- √72 = √36 × √2
Result: 6√2 ≈ 8.485281374
Babylonian (Heron’s) method
gₙ₊₁ = (gₙ + x ÷ gₙ) ÷ 2
- g
- Current guess; each step roughly doubles the number of correct digits
Worked example
√72, starting from 36.
- (36 + 72 ÷ 36) ÷ 2 = 19
- (19 + 72 ÷ 19) ÷ 2 = 11.395
- → 8.857 → 8.4931 → 8.485285
Result: Converges to 8.485281374…
Square Roots of 1 to 20
| n | √n | n | √n |
|---|---|---|---|
| 1 | 1 | 11 | 3.3166 |
| 2 | 1.4142 | 12 | 3.4641 (2√3) |
| 3 | 1.7321 | 13 | 3.6056 |
| 4 | 2 | 14 | 3.7417 |
| 5 | 2.2361 | 15 | 3.8730 |
| 6 | 2.4495 | 16 | 4 |
| 7 | 2.6458 | 17 | 4.1231 |
| 8 | 2.8284 (2√2) | 18 | 4.2426 (3√2) |
| 9 | 3 | 19 | 4.3589 |
| 10 | 3.1623 | 20 | 4.4721 (2√5) |
How to Read Your Result
Principal root and ±
The √ symbol means the principal (non-negative) root, so √25 = 5. When solving x² = 25, remember both +5 and −5 work; that is why equations often write x = ±√25.
Irrational roots
The square root of any whole number that is not a perfect square is irrational — its decimal never ends or repeats. The decimal shown is rounded; the simplified radical is exact, which is why math teachers prefer it.
Limitations & Accuracy Notes
- Decimals are double-precision floating point, accurate to about 15 significant figures.
- Radical simplification is done for whole numbers up to one trillion; decimals and fractions are shown as decimals only.
Frequently Asked Questions
How do you simplify a square root?
What is the square root of a negative number?
Why does a square root have two answers?
How do you find a square root by hand?
What are the perfect squares up to 400?
Is the square root of 2 a fraction?
References & Further Reading
- Methods of computing square roots — Wikipedia — Heron’s (Babylonian) method and convergence