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Square Root Calculator

Calculate square roots, cube roots, and nth roots of any number.

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

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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