Square Root Calculator
Calculate square roots, perfect squares, and simplify radical expressions instantly.
Perfect Squares Reference (1-20)
Quick Examples
Understanding Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √9 = 3 because 3 × 3 = 9.
Perfect squares are numbers whose square roots are whole numbers (1, 4, 9, 16, 25...).
Simplifying radicals: √72 can be written as √(36 × 2) = √36 × √2 = 6√2
About This Calculator
Ever wondered what number multiplied by itself gives you 49? That's exactly what a square root tells you - in this case, 7. Square roots are one of the most fundamental operations in mathematics, appearing everywhere from calculating distances to understanding compound interest.
Our Square Root Calculator shows you whether your input is a perfect square, simplifies radical expressions into their cleanest form, and walks you through every step. Think of square roots as the 'undo button' for squaring numbers - they help us reverse-engineer the original value from a squared result.
How to Use the Square Root Calculator
- 1**Choose Calculation Type**: Select Square Root (√x) or Square (x²)
- 2**Enter Your Number**: Type any positive number - whole or decimal
- 3**Adjust Precision**: Set decimal places (default: 6)
- 4**Review Results**: See if it is a perfect square and simplified form
- 5**Study Steps**: View the step-by-step calculation breakdown
Understanding Square Roots
A square root is a number that produces a specified value when multiplied by itself. √25 = 5 because 5 × 5 = 25.
| Perfect Square | Square Root |
|---|---|
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 49 | 7 |
| 64 | 8 |
| 81 | 9 |
| 100 | 10 |
Simplifying Radicals
To simplify √72: find the largest perfect square factor (36), then √72 = √36 × √2 = 6√2.
Real-World Applications
Square roots are used in the Pythagorean theorem, physics (pendulum periods), finance (standard deviation), and computer graphics (distance calculations).
Pro Tips
- 💡Memorize perfect squares up to 225 for faster mental math
- 💡Look for the LARGEST perfect square factor when simplifying
- 💡Use √(a × b) = √a × √b to break down complex calculations
- 💡Perfect squares mod 4 can only be 0 or 1
Frequently Asked Questions
In real numbers, negative numbers have no real square roots. In complex math, √(-1) = i.
