Square / Nth Root
Roots and radicals
About This Calculator
A root is the inverse operation of raising a number to a power. The square root finds what number multiplied by itself equals x, while the nth root finds the number that must be multiplied n times to reach x. Roots are fundamental in geometry, physics, and algebra.
Formula
Square root: sqrt(x) = x^(1/2)
Cube root: cbrt(x) = x^(1/3)
Nth root: n-th root of x = x^(1/n)
Example Calculation
Find the square root of 144 and the cube root of 27.
- sqrt(144) = 144^(1/2) = 12 (because 12 × 12 = 144)
- cbrt(27) = 27^(1/3) = 3 (because 3 × 3 × 3 = 27)
sqrt(144) = 12; cbrt(27) = 3
Perfect Squares 1 to 15
| n | n² | sqrt(n²) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
Frequently Asked Questions
What is the square root of a negative number?
In real numbers, the square root of a negative number is undefined. In complex numbers, sqrt(−1) = i, where i is the imaginary unit.
Is every number a perfect square?
No. Perfect squares are integers whose square root is also an integer (1, 4, 9, 16, 25, ...). All other positive numbers have irrational square roots.
How do I estimate a square root without a calculator?
Find the two nearest perfect squares. For sqrt(50): 7²=49 and 8²=64, so the answer is between 7 and 8 — closer to 7.07.