Square Root Calculator - √x Radical Calculator Square Root Calculator
Enter a number to instantly find its square root, check if it's a perfect square, and see the simplified radical form.
Enter a Number
Your Results
√144 12
Verification 12 × 12 = 144
Square Root 12
Perfect Square? Yes
Simplified Radical 12
Rounded (4 decimals) 12
Input Number 144
Root Index 2
Nearby Perfect Squares
121 (11²), 144 (12²), 169 (13²)
Square Root Calculator - Guide
What is a Square Root?
The square root of a number x is a value y such that y × y = x. It is written as √x or x^(1/2). For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots: a positive and a negative (e.g., √25 = ±5), but conventionally the positive root is used.
Key Concepts
1. Perfect Squares
Numbers whose square root is a whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...
2. Simplified Radical Form
√50 = √(25 × 2) = 5√2. Factor out perfect square components under the radical.
3. Nth Roots
The nth root of x is x^(1/n). Cube root: ∛8 = 2, Fourth root: ⁴√16 = 2.
Examples
- √144: 12 × 12 = 144, so √144 = 12 (perfect square)
- √50: √(25 × 2) = 5√2 ≈ 7.0711
- √2: An irrational number ≈ 1.4142
- ∛27: 3 × 3 × 3 = 27, so ∛27 = 3
Common Uses
- Geometry: Finding side lengths using the Pythagorean theorem (c = √(a² + b²)).
- Statistics: Calculating standard deviation involves square roots.
- Physics: Many physical formulas involve roots (e.g., velocity, energy).
- Finance: Compound growth rates and volatility calculations use roots.
How This Calculator Works
- Enter a number: Type any non-negative number.
- Choose root index: Default is 2 (square root). Change to 3 for cube root, etc.
- Instant results: Square root, perfect square check, simplified radical form, and nearby perfect squares update in real time.
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