MyCalcBuddy

Square Root Calculator

Calculate square roots, cube roots, and nth roots with exact and decimal answers. Shows the calculation method and works for any positive number or expression.

Enter Number

Square Root of 144

12.000000

Perfect Square

√Square Root
12.0000
∛Cube Root
5.2415
ⁿ√2th Root
12.0000
x²Squared
20,736

Common Roots

NumberSquare Root
11
42
93
164
255
366
497
648
819
10010
<>

Calculator Created & Verified By

Aleph Sterling

Lead Developer, MyCalcBuddy

📚

Formula Source: Handbook of Mathematical Functions

by Abramowitz & Stegun

🔄Last reviewed: May 2026
✓Formula accuracy verified against standard references

Transparency Note: "Aleph Sterling" is a pen name. While I maintain privacy, all formulas are sourced from verified references and cross-checked for accuracy. No credentials are claimed - only cited sources.

What is a Square Root?

The square root of a number is a value that, when multiplied by itself, gives the original number. Square roots are one of the most fundamental concepts in mathematics, appearing in geometry, algebra, physics, and engineering.

Definition:

  • If x² = n, then x = √n
  • √9 = 3 because 3 × 3 = 9
  • Every positive number has two square roots: +√n and -√n
  • The principal (positive) square root is denoted √n

Perfect Squares:

Number Square Root Number Square Root
1 1 49 7
4 2 64 8
9 3 81 9
16 4 100 10
25 5 121 11
36 6 144 12

Properties of Square Roots

Square roots follow several important mathematical properties:

Square Root Properties

Product Rule: √(a × b) = √a × √b Quotient Rule: √(a/b) = √a / √b Power Rule: (√a)² = a Absolute Value: √(a²) = |a| Nth Root: ⁿ√a = a^(1/n) Square root as exponent: √a = a^(1/2) Important Values: √2 ≈ 1.41421 √3 ≈ 1.73205 √5 ≈ 2.23607 √10 ≈ 3.16228

Where:

  • √= Square root symbol (radical)
  • ⁿ√= nth root
  • a, b= Non-negative real numbers

Simplifying Square Roots

A square root is simplified when no perfect square factors remain under the radical:

Steps to Simplify:

  1. Factor the number to find perfect square factors
  2. Apply the product rule: √(a×b) = √a × √b
  3. Take the square root of perfect squares
  4. Multiply any whole numbers outside the radical

Examples:

Original Factoring Simplified
√18 √(9 × 2) = √9 × √2 3√2
√50 √(25 × 2) = √25 × √2 5√2
√72 √(36 × 2) = √36 × √2 6√2
√200 √(100 × 2) = √100 × √2 10√2

How to Use This Square Root Calculator

Our calculator provides complete square root analysis:

  1. Enter Number: Input any positive real number
  2. Click Calculate: Get instant results
  3. View Results:
    • Exact decimal value
    • Simplified radical form (if applicable)
    • Whether it's a perfect square
    • Prime factorization

Additional Features:

  • Calculate any nth root (cube root, 4th root, etc.)
  • Simplify radical expressions
  • Handle fractions and decimals
  • Show step-by-step simplification

Input Options:

  • Whole numbers: 50, 200, 1000
  • Decimals: 2.5, 0.16
  • Fractions: 1/4, 9/16
  • Expressions: √(a/b)

Irrational Numbers and Square Roots

Most square roots are irrational numbers - they cannot be expressed as exact fractions and have infinite, non-repeating decimal expansions:

Rational vs Irrational Square Roots:

  • Rational: √n is rational only if n is a perfect square (1, 4, 9, 16, 25...)
  • Irrational: √2, √3, √5, √7, √11... all non-perfect-square roots

Famous Irrational Square Roots:

  • √2 ≈ 1.41421356... - diagonal of unit square
  • √3 ≈ 1.73205080... - height of equilateral triangle with side 2
  • √5 ≈ 2.23606797... - related to golden ratio (φ = (1+√5)/2)

Proof that √2 is Irrational:

If √2 = a/b (reduced fraction), then 2 = a²/b², so a² = 2b². This means a² is even, so a is even. Write a = 2k. Then 4k² = 2b², so b² = 2k², meaning b is also even. But if both a and b are even, a/b wasn't reduced - contradiction! Therefore √2 cannot be a fraction.

Complex Numbers and Negative Square Roots

In real numbers, square roots of negative numbers are undefined. However, complex numbers extend this:

Imaginary Numbers

Definition: i = √(-1) i² = -1 i³ = -i i⁴ = 1 (cycle repeats) Square root of negative number: √(-n) = √n × √(-1) = √n × i = i√n Examples: √(-1) = i √(-4) = 2i √(-9) = 3i √(-2) = i√2 ≈ 1.414i

Where:

  • i= Imaginary unit (√-1)
  • a + bi= Complex number format

Applications of Square Roots

Square roots appear throughout mathematics, science, and everyday life:

Geometry:

  • Pythagorean Theorem: c = √(a² + b²) for right triangles
  • Distance formula: d = √((x₂-x₁)² + (y₂-y₁)²)
  • Circle: r = √(A/π) to find radius from area
  • Diagonal of square: d = s√2

Physics:

  • Velocity: v = √(2gh) for falling objects
  • Wave speed: Related to √(tension/density)
  • Pendulum period: T = 2π√(L/g)

Statistics:

  • Standard deviation: σ = √(variance)
  • Standard error: SE = σ/√n
  • Root Mean Square (RMS)

Finance:

  • Volatility calculations
  • Option pricing models
  • Risk assessment

Worked Examples

Simplify a Square Root

Problem:

Simplify √72

Solution Steps:

  1. 1Find the largest perfect square factor of 72
  2. 272 = 36 × 2 (36 is perfect square)
  3. 3Apply product rule: √72 = √36 × √2
  4. 4√36 = 6
  5. 5Result: √72 = 6√2
  6. 6Decimal check: 6 × 1.414 ≈ 8.485 ≈ √72 ✓

Result:

√72 = 6√2 ≈ 8.485

Using Pythagorean Theorem

Problem:

Find the hypotenuse of a right triangle with legs 5 and 12

Solution Steps:

  1. 1Use Pythagorean theorem: c² = a² + b²
  2. 2c² = 5² + 12²
  3. 3c² = 25 + 144 = 169
  4. 4c = √169
  5. 5169 is a perfect square (13²)
  6. 6c = 13

Result:

Hypotenuse = 13

Distance Between Points

Problem:

Find distance between (1, 2) and (4, 6)

Solution Steps:

  1. 1Use distance formula: d = √((x₂-x₁)² + (y₂-y₁)²)
  2. 2d = √((4-1)² + (6-2)²)
  3. 3d = √(3² + 4²)
  4. 4d = √(9 + 16)
  5. 5d = √25 = 5

Result:

Distance = 5 units

Tips & Best Practices

  • ✓Memorize perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
  • ✓To simplify √n, find the largest perfect square factor
  • ✓√(a²) = |a| (always positive), not just a
  • ✓Common approximations: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236
  • ✓Square root of a fraction: √(a/b) = √a/√b
  • ✓Check your simplification by squaring your answer
  • ✓For quick estimation: √n lies between √(perfect square below n) and √(perfect square above n)

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Frequently Asked Questions

Because any real number squared is positive (or zero). Positive × positive = positive, and negative × negative = positive. There's no real number that, when multiplied by itself, gives a negative result. To handle √(-1), mathematicians invented imaginary numbers, where i = √(-1). This extends our number system to complex numbers (a + bi).
Because both a positive and negative number, when squared, give the same positive result. For example, both 3² = 9 and (-3)² = 9. So technically, the square roots of 9 are +3 and -3. When we write √9, we mean the principal (positive) square root, which is 3. In equations like x² = 9, we write x = ±3 to include both.
A square root √n is rational if and only if n is a perfect square (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...). You can check by finding the prime factorization: if every prime factor appears an even number of times, it's a perfect square. For example, 36 = 2² × 3², so √36 = 6 is rational.
They're the same thing - both mean square root. The small 2 is usually omitted because square root is the default. For other roots, we write ³√ (cube root), ⁴√ (fourth root), etc. These are called 'nth roots' and ⁿ√a = a^(1/n). Odd roots can be taken of negative numbers: ³√(-8) = -2.
Several methods exist: (1) Factor into perfect squares: √50 = √(25×2) = 5√2. (2) Use long division algorithm (similar to division). (3) Newton's method (guess and iterate): start with estimate x, improve with (x + n/x)/2. (4) For estimation, find surrounding perfect squares: √50 is between √49=7 and √64=8.
The radical is the √ symbol. The number under it is called the 'radicand.' In √50, the radical is √ and the radicand is 50. In ³√27, it's a cube root radical with radicand 27. Numbers outside the radical (like 3 in 3√2) are called 'coefficients' of the radical expression.

Sources & References

Last updated: 2026-01-22

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