Square Root Calculator - √ Cube Root & nth Root

√ Root Calculator

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√144

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.

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!

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!

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.

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

The square root of a number x is a value y such that y² = x. For example, √25 = 5 because 5² = 25. Every positive number has two square roots: a positive principal root and a negative root. The square root of 0 is 0. The square root of a negative number is not a real number - it is imaginary (written using the imaginary unit i).

A perfect square is an integer that is the exact square of another integer, so its square root comes out as a whole number with no decimal. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400… √64 = 8 exactly. √65 ≈ 8.0622… is irrational. Use the Perfect Squares tab above to browse all squares from 1² to 30².

Find the largest perfect square factor of the number, then pull it outside the radical. √72 = √(36×2) = 6√2. √48 = √(16×3) = 4√3. √200 = √(100×2) = 10√2. √98 = √(49×2) = 7√2. The rule that makes this work: √(a×b) = √a × √b. Remember: √(a+b) ≠ √a + √b - that's a very common algebra mistake.

Not in real numbers - the result is undefined. But in complex mathematics, √(−1) = i (imaginary unit). So √(−9) = 3i, √(−16) = 4i. Complex numbers of the form a + bi are not just theoretical - they are essential in electrical engineering (AC circuits use them directly), quantum mechanics, and signal processing. The calculator shows the imaginary form for negative inputs.

The cube root of x (∛x) is the value y such that y³ = x. √25 = 5 because 5² = 25. ∛27 = 3 because 3³ = 27. Key difference: cube roots can be negative - ∛(−8) = −2 because (−2)³ = −8. Square roots of negative numbers are not real. Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000…

The nth root of x is the value y such that yⁿ = x, written as ⁿ√x or x^(1/n). The 4th root of 81 = 81^(1/4) = 3 because 3⁴ = 81. The 5th root of 32 = 32^(1/5) = 2 because 2⁵ = 32. Even-indexed roots (4th, 6th…) of negative numbers are not real. Odd-indexed roots (3rd, 5th…) of negative numbers are real and negative. This calculator handles any root index from 2 to 100.

The Babylonian method (also called Newton's method) is an ancient iterative algorithm to approximate √N. Start with any guess g. Then repeatedly compute: g = (g + N/g) / 2. Each step roughly doubles the number of correct decimal places. For √2 starting at g = 1: Step 1: g = (1 + 2/1)/2 = 1.5. Step 2: g = (1.5 + 2/1.5)/2 = 1.4167. Step 3: g = 1.41422. Correct to 5 places in just 3 steps. It's the same algorithm your calculator chip essentially uses.

√2 = 1.41421356237… and its decimal digits never end, never repeat, and never show any pattern - this is what 'irrational' means. It was the first number ever proven to be irrational, around 500 BCE by the ancient Greeks, which caused a genuine philosophical crisis at the time. It matters because √2 appears in the diagonal of any unit square (the 1×1×√2 right triangle), in A-series paper dimensions (A4 is always 1:√2), in music tuning (each semitone in equal temperament is a ratio of ²¹²√2), and throughout mathematics and physics.

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)

How this calculator works, and where the numbers come from

The Square Root Calculator applies the standard formula for this calculation to the values you enter and updates the result as you type. The calculation itself happens in your browser, and the page explains the method so you can check any result by hand.

Please note: Results are provided for general information and are calculated from the values you enter.

Sources and further reading

Last reviewed: October 6, 2026 by the CalcQube Editorial Team. See our editorial policy for how we build and check calculators, or report an error.

√ Square Root Calculator