Square Root Calculator
Calculate the square root, cube root, or any nth root of a number. Enter a number and the root to compute the exact decimal result and simplified radical form.
Perfect Squares Reference
| Number | Square Root | Perfect Square? |
|---|
Frequently Asked Questions
A perfect square is a number whose square root is a whole number. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Not in the real number system. The square root of a negative number is imaginary — written as i√|n|. For example, √(-9) = 3i.
What a square root is
The square root of a number is the value that, multiplied by itself, gives that number. Since 7 × 7 = 49, the square root of 49 is 7.
Strictly, every positive number has two square roots — 7 and −7 both square to 49. The √ symbol refers to the principal (positive) root, which is what this calculator returns.
Perfect squares
These come out as whole numbers and are worth recognising on sight:
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 9 | 81 |
| 2 | 4 | 10 | 100 |
| 3 | 9 | 11 | 121 |
| 4 | 16 | 12 | 144 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 20 | 400 |
| 7 | 49 | 25 | 625 |
| 8 | 64 | 30 | 900 |
Estimating without a calculator
Find the perfect squares either side and interpolate.
Example: √70. It sits between 64 (8²) and 81 (9²), so the answer is between 8 and 9. 70 is closer to 64, so estimate around 8.3. The actual value is 8.3666 — close enough for a sanity check.
Irrational results
Most square roots are irrational: the decimal runs forever without repeating. √2 = 1.41421356… and never resolves. Any displayed value is rounded, which is why exact-form answers like 5√2 are preferred in algebra over 7.071.
Negative numbers
No real number squares to a negative, because a negative times a negative is positive. √−9 has no real solution. In complex arithmetic it is written 3i, where i is defined as √−1 — a concept you'll meet in engineering and advanced algebra, but outside the scope of real-number calculation.
Where it's used
Finding a side length from an area, the Pythagorean theorem, standard deviation in statistics, the quadratic formula, and distance between two points.