Loading Calculator...
Please wait a moment
Please wait a moment
Calculate square roots with both exact and decimal forms. Simplify radicals, check for perfect squares, and see step-by-step solutions.
Enter a number to calculate its square root
Type any non-negative number in the input field. You can use whole numbers, decimals, or fractions.
The calculator shows both the simplified radical form and decimal approximation. Perfect squares are automatically identified.
Review the step-by-step solution to see how the radical was simplified using prime factorization.
√x = y where y² = x
The square root of x is the number that, when multiplied by itself, gives x.
√(ab) = √a × √b
Example: √72 = √(36×2) = √36 × √2 = 6√2
√(a/b) = √a / √b
Example: √(25/4) = √25 / √4 = 5/2
√(x²) = |x|
Example: √((-5)²) = √25 = 5
| Number | Square | Square Root |
|---|---|---|
| 1 | 1 | √1 = 1 |
| 2 | 4 | √4 = 2 |
| 3 | 9 | √9 = 3 |
| 4 | 16 | √16 = 4 |
| 5 | 25 | √25 = 5 |
| 6 | 36 | √36 = 6 |
| 7 | 49 | √49 = 7 |
| 8 | 64 | √64 = 8 |
| 9 | 81 | √81 = 9 |
| 10 | 100 | √100 = 10 |
| 11 | 121 | √121 = 11 |
| 12 | 144 | √144 = 12 |
| 13 | 169 | √169 = 13 |
| 14 | 196 | √196 = 14 |
| 15 | 225 | √225 = 15 |
A square root of a number x is a value that, when multiplied by itself, gives x. For example, the square root of 25 is 5 because 5 × 5 = 25. It's denoted by the radical symbol √.
To simplify √x, find the prime factorization of x. For each pair of identical factors, take one factor outside the radical. For example, √72 = √(2×2×2×3×3) = 2×3√2 = 6√2.
A perfect square is a number that is the square of an integer. Examples include 1, 4, 9, 16, 25, 36, etc. Perfect squares have exact integer square roots (e.g., √16 = 4 exactly).
In real numbers, you cannot take the square root of a negative number. However, in complex numbers, √(-1) is defined as i (the imaginary unit). For example, √(-4) = 2i.
The symbol √4 (called the principal square root) always refers to the positive root: √4 = 2. The notation ±√4 means both the positive and negative roots: +2 and -2. This distinction is important when solving equations.
Find the two perfect squares that your number falls between. For example, to estimate √50, note that 49 < 50 < 64, so √49 < √50 < √64, meaning 7 < √50 < 8. Since 50 is closer to 49, √50 ≈ 7.1.
Square roots are one of the most fundamental operations in mathematics. The square root operation is the inverse of squaring a number. While squaring asks "what do I get when I multiply this number by itself?", the square root asks "what number, when multiplied by itself, gives me this?"
Square roots appear throughout mathematics and science: in the Pythagorean theorem for finding distances, in quadratic equations, in statistics for standard deviation, and in physics for calculating velocities and energies.