Math / Square Root Calculator
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√ Square Root Calculator

Calculate square roots, cube roots, and nth roots instantly with full step-by-step workings. See whether a number is a perfect square, get the simplified radical form, check the perfect squares and cubes tables, and understand the Babylonian method - all in one free tool.

√ Root Calculator

√144
12

📋 Perfect Squares (1–30)

Click any card to calculate its square root.

🎯 Perfect Cubes (1–20)

Click any card to calculate its cube root.

💡 Square Root Facts & History

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Ancient Origins
Square roots were known to the Babylonians around 1800 BCE. The Yale Babylonian Collection contains a clay tablet (YBC 7289) showing the square root of 2 calculated to an extraordinary 6 decimal places - a remarkable feat for the era!
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√2 - The First Irrational Number
The Pythagoreans discovered that √2 cannot be expressed as a fraction. This caused a crisis in ancient Greek mathematics - they believed all numbers were rational. Legend says Hippasus, who proved this, was thrown overboard at sea by fellow Pythagoreans!
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√2 Never Ends
√2 = 1.41421356237... and the digits never repeat or end - it's irrational. It has been calculated to over 10 trillion decimal places. Despite this, just 39 digits of √2 is enough to compute the circumference of the observable universe to an accuracy within a hydrogen atom!
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The Golden Ratio Connection
The golden ratio φ = (1 + √5) / 2 ≈ 1.6180... involves a square root. It appears everywhere in nature - sunflower seeds, nautilus shells, galaxy spirals - and was used in the Parthenon and the Great Pyramid's proportions.
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Pythagorean Theorem
The most famous use of square roots: c = √(a² + b²). If a right triangle has legs 3 and 4, the hypotenuse is √(9+16) = √25 = 5. The 3-4-5 triangle was used by Egyptian rope-stretchers to make perfect right angles for pyramid construction!
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Fast Inverse Square Root
The game Quake III (1999) contained a legendary hack: the "Fast Inverse Square Root" - a 4-line code that calculated 1/√x using a mysterious magic number (0x5f3759df). It was 4× faster than the standard method and transformed 3D gaming. The code is still studied today!
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Square Root in Nature
The standard deviation in statistics uses square roots. The distance formula uses square roots. Even Einstein's relativity (Lorentz factor γ = 1/√(1-v²/c²)) uses square roots. They're fundamental to describing how the physical universe works.
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Negative Square Roots & Imaginary Numbers
The square root of a negative number is called imaginary. √(-1) = i (imaginary unit). Complex numbers (a + bi) use this concept and are essential in electrical engineering, quantum mechanics, and signal processing. They're not imaginary in the real-world sense!

📐 Root Formulas & Methods

Basic Definitions

Square Root: √x = x^(1/2) → if √x = y, then y² = x
Cube Root: ∛x = x^(1/3) → if ∛x = y, then y³ = x
Nth Root: ⁿ√x = x^(1/n) → if ⁿ√x = y, then yⁿ = x

Properties of Square Roots

√(a × b) = √a × √b
√(a / b) = √a / √b
√(a²) = |a| (absolute value)
(√a)² = a (for a ≥ 0)
√a + √b ≠ √(a+b) ← Common mistake!

Simplifying Square Roots

Find the largest perfect square factor and pull it out.

√72 = √(36 × 2) = √36 × √2 = 6√2
√48 = √(16 × 3) = √16 × √3 = 4√3
√200 = √(100 × 2) = 10√2

Babylonian Method (Newton's Method)

Ancient iterative algorithm to calculate √N:

Start: guess g = N/2
Repeat: g = (g + N/g) / 2
Until g² ≈ N

Example: √2, start g=1
g₁ = (1 + 2/1) / 2 = 1.5
g₂ = (1.5 + 2/1.5) / 2 = 1.4167
g₃ = (1.4167 + 2/1.4167) / 2 = 1.41422...
Converges very fast!

Perfect Square Check

A number n is a perfect square if:
√n is an integer (no decimal part)
OR: n = k² for some integer k

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

Distance Formula (Uses √)

Distance = √((x₂-x₁)² + (y₂-y₁)²)

Example: Distance from (0,0) to (3,4):
= √((3-0)² + (4-0)²)
= √(9 + 16) = √25 = 5

❓ Frequently Asked Questions

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⚡ Common Square Roots

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Square Root Calculator - Everything You Need to Know About Roots

Square roots show up everywhere - in geometry, physics, statistics, finance, and everyday calculations. Whether you need a quick answer, a step-by-step explanation for a homework problem, or a deep understanding of why irrational numbers exist, this calculator covers all of it. Enter any number and instantly see the result, whether it's a perfect square, its simplified radical form, and a worked solution using the Babylonian method.

What this calculator handles: Square roots (√x) · Cube roots (∛x) · Nth roots (ⁿ√x for any n from 2–100) · Squaring a number (x²) · Cubing a number (x³) · Step-by-step solutions · Perfect square detection · Simplified radical form · Related calculations

Square Root - The Basics

The square root of a number x is the value y that satisfies y² = x. The symbol √ is called a radical sign. Every positive number has two square roots - a positive one (called the principal root) and a negative one - but when someone says "the square root of 9," they mean the positive root, which is 3.

  • √0 = 0 - zero is its own square root
  • √1 = 1 - one is also its own square root
  • √144 = 12 - perfect square, exact whole number answer
  • √2 ≈ 1.41421356… - irrational, the decimal never ends or repeats
  • √(−4) - not a real number; result is 2i in complex mathematics

Perfect Squares vs Irrational Roots

This is one of the most important distinctions in basic math. A perfect square is any number that is the square of a whole integer - its square root comes out as a clean, exact integer. Every other positive number has an irrational square root - the decimal goes on forever with no repeating pattern.

✓ Perfect Squares (exact integer roots)

  • √1 = 1, √4 = 2, √9 = 3, √16 = 4
  • √25 = 5, √36 = 6, √49 = 7, √64 = 8
  • √81 = 9, √100 = 10, √121 = 11, √144 = 12
  • √169 = 13, √196 = 14, √225 = 15, √256 = 16
  • √289 = 17, √324 = 18, √361 = 19, √400 = 20

~ Irrational Roots (decimal, never ends)

  • √2 ≈ 1.41421356…
  • √3 ≈ 1.73205080…
  • √5 ≈ 2.23606797…
  • √6 ≈ 2.44948974…
  • √7 ≈ 2.64575131…
  • √8 ≈ 2.82842712… (= 2√2)
  • √10 ≈ 3.16227766…
  • √12 ≈ 3.46410161… (= 2√3)

How to Simplify a Square Root (Simplest Radical Form)

When a square root is not a perfect square, the standard approach is to simplify it into its simplest radical form by factoring out any perfect square hiding inside the number:

  • √72 → find largest perfect square factor: 36 → √(36 × 2) = 6√2
  • √48 → largest perfect square factor: 16 → √(16 × 3) = 4√3
  • √200 → largest perfect square factor: 100 → √(100 × 2) = 10√2
  • √50 → largest perfect square factor: 25 → √(25 × 2) = 5√2
  • √98 → largest perfect square factor: 49 → √(49 × 2) = 7√2

The key rule to remember: √(a × b) = √a × √b. This is valid. But √(a + b) ≠ √a + √b - that's one of the most common mistakes in algebra.

Cube Roots and Nth Roots

The cube root of x (written ∛x) is the value y such that y³ = x. Unlike square roots, cube roots can be negative - ∛(−8) = −2 because (−2)³ = −8. The nth root extends this: ⁿ√x = x^(1/n), so the 4th root of 81 is 81^(1/4) = 3 because 3⁴ = 81.

  • Even roots (4th, 6th…) of negative numbers are not real
  • Odd roots (3rd, 5th…) of negative numbers are real and negative
  • The calculator handles nth roots for any n from 2 to 100

Where Square Roots Actually Show Up in Real Life

Square roots aren't just a school topic - they're built into formulas used in science, engineering, and finance every day:

  • Pythagorean theorem: c = √(a² + b²) - used in construction, navigation, and screen diagonal measurements (a 16×9 screen has diagonal √(16²+9²) = √337 ≈ 18.4 units)
  • Standard deviation in statistics: σ = √(variance) - used in finance, science, and quality control
  • Distance formula: d = √((x₂−x₁)² + (y₂−y₁)²) - used in mapping and GPS
  • Physics - velocity from kinetic energy: v = √(2KE/m)
  • Finance - volatility: annualised volatility = daily volatility × √252 (trading days per year)
  • Einstein's relativity: Lorentz factor γ = 1/√(1−v²/c²)