Square Root Calculator
Calculate Square Roots
Find principal square roots, simplify radicals, and get exact or approximate solutions.
Square Root Result
√25 = 5
Number Properties:
Radical Simplification:
Square Root Magnitude:
The square root of a number is the value that, when multiplied by itself, gives the original number.
What is a Square Root?
Square Root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25. The square root operation is the inverse of squaring a number.
Square Root Properties
Principal Square Root
Non-negative result
Standard definition
Negative Square Root
Negative solution
Valid for equations
Perfect Squares
Integer results
Exact values
Radical Form
Simplification rules
Exact expressions
Square Root Rules
1. Perfect Square Definition
A perfect square is a number that has an integer as its square root:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
2. Radical Simplification
Square roots can be simplified using the product property:
√(a × b) = √a × √b
3. Square Root of Fractions
The square root of a fraction equals the square root of numerator divided by square root of denominator:
√(a/b) = √a / √b
Real-World Applications
Geometry & Measurement
- Pythagorean theorem: Finding side lengths in right triangles
- Area calculations: Finding side length from area of squares
- Distance formula: Calculating distances between points
- Circle geometry: Radius from area or circumference
Physics & Engineering
- Kinematics: Calculating velocities and accelerations
- Electrical engineering: RMS (Root Mean Square) values
- Structural engineering: Stress and strain calculations
- Wave mechanics: Amplitude and energy relationships
Statistics & Data Analysis
- Standard deviation: Measuring data dispersion
- Variance: Square root relationship with standard deviation
- Normal distribution: Z-score calculations
- Error analysis: Root mean square error (RMSE)
Finance & Economics
- Compound interest: Time period calculations
- Risk assessment: Volatility measurements
- Portfolio theory: Standard deviation of returns
- Economic modeling: Growth rate calculations
Common Square Root Examples
| Number | Square Root | Type | Simplified Radical |
|---|---|---|---|
| 16 | 4 | Perfect Square | 4 |
| 25 | 5 | Perfect Square | 5 |
| 50 | ≈ 7.071 | Non-Perfect | 5√2 |
| 72 | ≈ 8.485 | Non-Perfect | 6√2 |
Perfect Squares Reference
| Number | Square | Square Root | Number | Square | Square Root |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 11 | 121 | 11 |
| 2 | 4 | √2 ≈ 1.414 | 12 | 144 | 12 |
| 3 | 9 | √3 ≈ 1.732 | 13 | 169 | √13 ≈ 3.606 |
| 4 | 16 | 4 | 14 | 196 | √14 ≈ 3.742 |
| 5 | 25 | 5 | 15 | 225 | √15 ≈ 3.873 |
Step-by-Step Calculation Process
Example 1: Perfect Square (√144)
- Identify the number: 144
- Check if it's a perfect square: 12 × 12 = 144
- Result: √144 = 12
- Verification: 12 × 12 = 144 ✓
Example 2: Non-Perfect Square (√50)
- Identify the number: 50
- Find perfect square factors: 50 = 25 × 2
- Apply radical rule: √50 = √(25 × 2) = √25 × √2
- Simplify: 5 × √2
- Decimal approximation: 5 × 1.414 = 7.07
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Frequently Asked Questions (FAQs)
Q: Can square roots be negative?
A: The principal square root is always non-negative. However, every positive number has two square roots: one positive and one negative. For example, both 5 and -5 are square roots of 25.
Q: What is the square root of 0?
A: The square root of 0 is 0, since 0 × 0 = 0.
Q: How do you simplify square roots?
A: Factor the number under the radical into perfect squares and other factors, then take the square root of the perfect squares outside the radical.
Q: What's the difference between exact and approximate square roots?
A: Exact square roots are expressed in radical form (like √2), while approximate square roots are decimal values (like 1.414). Perfect squares have exact integer square roots.
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