Square Root Calculator: Find Roots Instantly
Calculate the square root of any number instantly.
Square Root Calculator
Calculate the square root of any positive number
Tips:
What is a Square Root?
The square root of a number is a value that, when multiplied by itself, equals the original number. It is written using the √ symbol. For example, √64 = 8 because 8 × 8 = 64.
Common Perfect Squares (Quick Reference)
| Number (n) | n² | √(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 |
| 12 | 144 | 12 |
| 15 | 225 | 15 |
| 20 | 400 | 20 |
How to Calculate Square Roots Manually
- 1Check if the number is a perfect square (like 9, 16, 25, 36, 49, 64).
- 2If it is, find the integer that multiplies by itself to get the number (√64 = 8).
- 3If it’s not a perfect square, find two nearby perfect squares (e.g., 50 is between 49 and 64).
- 4Estimate the root between them (√50 is between 7 and 8).
- 5Use a calculator to get a precise decimal value if needed.
Where Square Roots Are Used
- Geometry: finding side lengths with the Pythagorean theorem
- Statistics: standard deviation and variance calculations
- Physics: speed, distance, and energy relationships
- Engineering: signal processing and root-mean-square (RMS) values
- Finance: volatility and risk calculations (e.g., square-root-of-time rule)
Quick tip: √x usually means the principal square root (the positive root). Even though both 5 and -5 square to 25, √25 is defined as 5 in most contexts.
Perfect Squares vs. Non-Perfect Squares
Perfect squares have whole-number square roots (like √81 = 9). Most numbers are not perfect squares, so their square roots are decimals (like √2 ≈ 1.414). Those decimal values can be rounded depending on your need (2 decimals, 4 decimals, etc.).
Visualizing Growth: n vs. n²
How Squaring Grows Faster Than n
n² · values shown as provided
Why √ Works (Square Area Intuition)
How to Use the Square Root Calculator
- 1Enter the number you want the square root of (example: 64).
- 2Click calculate to get the result instantly.
- 3If the number is not a perfect square, you’ll see a decimal approximation.
- 4Round the result to the precision you need (2 decimals, 4 decimals, etc.).
- 5Use nearby perfect squares to sanity-check the result.
Square Root Calculator FAQs
What is the square root of a number?
The square root of a number is a value that gives the original number when multiplied by itself. For example, √49 = 7 because 7 × 7 = 49.
What is a perfect square?
A perfect square is a number that can be written as n² for some integer n. Examples include 1, 4, 9, 16, 25, 36, 49, 64, and 100.
Can this calculator find decimal square roots?
Yes. If the number is not a perfect square (like 2 or 50), the calculator returns a decimal approximation.
Why does a number have two square roots?
Because both positive and negative values square to the same number. For example, 5² = 25 and (-5)² = 25. The √ symbol usually refers to the principal (positive) square root.
Can I calculate the square root of a negative number?
In real numbers, square roots of negative numbers are not defined. In complex numbers, √-1 is i, and √(-a) = i√a for positive a.
What is √0?
√0 = 0 because 0 × 0 = 0.
What’s the difference between √x and x²?
√x finds the number that squares to x, while x² squares x. They are inverse operations for x ≥ 0.
Is the square root calculator accurate?
Yes. It returns exact results for perfect squares and accurate decimal approximations for non-perfect squares.