What is a square root?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √144 = 12 because 12×12 = 144.

What is a perfect square?

A perfect square is a number whose square root is a whole number. Examples: 1, 4, 9, 16, 25, 36, etc.

Can you take the square root of a negative number?

In real numbers, you cannot take the square root of a negative number. Complex numbers (using i = √-1) are needed for that.

How is square root different from cube root?

Square root finds a value that squares to the original number, while cube root finds a value that cubes to the original number.

Math Calculators

Square Root Calculator

Calculate square roots, cube roots, and nth roots with high decimal precision for mathematical, scientific, and engineering applications. Features step-by-step root calculation methods showing work process, support for any nth root (4th, 5th, 10th, etc.), radical simplification for perfect squares, decimal and fractional output formats, negative number handling with imaginary results, and educational explanations perfect for algebra, geometry, calculus, physics, and construction calculations.

How to Use the Square Root Calculator

Use the Square Root Calculator to square roots, cube roots, and nth roots with high decimal precision for mathematical, scientific, and engineering applications. Features step-by-step root calculation methods showing work process, support for any nth root (4th, 5th, 10th, etc.), radical simplification for perfect squares, decimal and fractional output formats, negative number handling with imaginary results, and educational explanations perfect for algebra, geometry, calculus, physics, and construction calculations.. Enter your values to get accurate, instant results tailored to your situation.

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Calculate square accurately and instantly
Calculate root accurately and instantly
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Calculate square accurately and instantly
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Square Root Guide

Radical operations

Expert Tips

Square Root Undoes Squaring: √(x²) = x: √16 = 4 because 4² = 16. √49 = 7 because 7² = 49. √2 ≈ 1.414 (irrational). Negative square roots don't exist in real numbers (√-16 = imaginary 4i).
Perfect Squares Worth Memorizing: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100, 11²=121, 12²=144, 15²=225, 20²=400. Know these for instant recognition and mental math.
Simplify by Factoring Perfect Squares: √72 = √(36×2) = √36 × √2 = 6√2. √50 = √(25×2) = 5√2. √200 = √(100×2) = 10√2. Factor out largest perfect square for simplified radical form.
Estimation Technique for Non-Perfect Squares: √50: Nearest perfect squares 49 (7²) and 64 (8²). √50 between 7 and 8, closer to 7. Actual ≈ 7.07. √80: Between 64 (8²) and 81 (9²), closer to 9 → ≈ 8.9. Useful for mental approximation.

Essential Fundamentals — Square root basics

Properties

Product Property: √(a×b) = √a × √b. Example: √12 = √(4×3) = √4 × √3 = 2√3. Enables simplification. Reverse: √2 × √8 = √16 = 4.
Quotient Property: √(a/b) = √a / √b. Example: √(9/4) = √9 / √4 = 3/2. Rationalize denominators: 1/√2 = 1/√2 × √2/√2 = √2/2. Avoids radicals in denominator.

Advanced Strategies — Simplification techniques

Radical Operations

Rationalizing Denominators: Convert 5/√3 to rational denominator: multiply by √3/√3 → 5√3/3. For √a+√b denominators, use conjugate: 1/(√5+√3) × (√5-√3)/(√5-√3) = (√5-√3)/2. Standard form for final answers.
Nested Radicals: Simplify √(√16) = √4 = 2. Or √(3+2√2): Look for pattern (a+b)² = a²+2ab+b². Trial: (1+√2)² = 1+2√2+2 = 3+2√2 ✓. So √(3+2√2) = 1+√2. Advanced technique for special cases.

Frequently Asked Questions

What is a square root?: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √144 = 12 because 12×12 = 144.
What is a perfect square?: A perfect square is a number whose square root is a whole number. Examples: 1, 4, 9, 16, 25, 36, etc.
Can you take the square root of a negative number?: In real numbers, you cannot take the square root of a negative number. Complex numbers (using i = √-1) are needed for that.
How is square root different from cube root?: Square root finds a value that squares to the original number, while cube root finds a value that cubes to the original number.