SamCalculator

Square Root Calculator

Calculate square roots instantly with step-by-step solutions, radical simplification, perfect square detection, and decimal approximations.

Square Root √x

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What Is a Square Root?

The square root of a non-negative number x — written √x — is the value that, multiplied by itself, returns x. √144 = 12 because 12 × 12 = 144; √50 = 5√2 because 50 = 25 × 2 and √25 = 5; √2 ≈ 1.41421356 because √2 is irrational and doesn't reduce to a clean integer or fraction. The square root is the inverse operation of squaring, and it sits underneath a huge chunk of algebra, geometry, statistics, and physics.

This calculator pairs the three most-asked root tools on one page: a Square Root Calculator with radical simplification and perfect-square detection, a Cube Root Calculator that handles positive, negative, and decimal inputs, and a General nth-Root Calculator for fourth, fifth, sixth, and any custom-index root. It works alongside the scientific calculator, the Pythagorean theorem calculator, and the percentage calculator.

How the Calculator Works

Read the radicand

The number under the radical is called the radicand. The calculator accepts any real number — positive, negative, integer, or decimal — and validates it as you type.

Check for a perfect square

Before computing, the tool tests whether the radicand is a perfect square (1, 4, 9, 16, 25, 36, …). If it is, the answer is an exact integer with no decimal approximation needed.

Simplify the radical form

If the radicand isn't a perfect square, the calculator factors out the largest square divisor and writes the answer as k√m where m is square-free — e.g. √50 → 5√2, √72 → 6√2, √200 → 10√2.

Compute and verify

A high-precision decimal approximation (~15 significant figures) is shown alongside the simplified form, and a verification line squares the answer back to confirm.

6 Ways to Use This Calculator

1

Find a clean square root

Drop in a perfect square — 81, 144, 256, 529, 1024 — and the tool returns the exact integer with verification.

2

Simplify any radical

Enter 50, 75, 200, or 288 to see the simplified k√m form (5√2, 5√3, 10√2, 12√2). Saves time on algebra homework and helps recognise equivalent radicals.

3

Get a decimal approximation

For irrational radicands like √2, √3, √5, or √π the tool returns a high-precision decimal you can paste straight into a spreadsheet or report.

4

Compute a cube root

Switch to the Cube tab for ∛x problems including ∛(−8) = −2, ∛125 = 5, and arbitrary decimal radicands.

5

Take any nth root

The General tab solves ⁴√16, ⁵√32, ⁶√64, ⁷√128, or any custom-index root — with a built-in domain check for even roots of negative numbers.

6

Read the step-by-step

Every result includes a worked solution — factoring, applying the formula, computing the decimal, and verifying — so the calculator doubles as a learning tool.

Best Practices

Recognise the small perfect squares. Memorising 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 makes mental simplification fast — when the radicand starts with one of these as a factor, the radical simplifies cleanly.

Carry exact and decimal forms together. Exact form (e.g. 5√2) preserves the algebra; the decimal (≈ 7.071) is what an engineer or surveyor uses on the ground. The calculator returns both — pick the one that matches your context.

Take negatives seriously. The square root of a negative number isn't real — it's imaginary, written with the unit i = √−1. The calculator handles this case by returning the imaginary form (e.g. √−4 = 2i) instead of silently flipping the sign.

Why Square Roots Matter

Geometry & distance

The Pythagorean theorem turns into a square-root operation any time you want a side length out of the squared-sum a² + b². Every distance formula, every diagonal calculation, every shortest-path measurement is a square root.

Statistics & data science

Standard deviation is the square root of variance. Confidence intervals scale with √n. Whenever spread or noise is involved, square roots appear — they damp big numbers into intuitively human-scale ones.

Physics & engineering

Pendulum period (T = 2π√(L/g)), terminal velocity, RMS voltage, kinetic energy → speed conversion — all use square roots to invert squared quantities back to linear ones.

Construction & finance

Carpenters use √(a² + b²) for diagonals; financial analysts use √(Σweights × σ²) for portfolio risk. The same operation underlies practical work in trades and capital markets.

Tricky Cases

Square root of zero

√0 = 0 exactly. Zero is its own square root because 0 × 0 = 0, and it's the only number with just one real square root.

Negative radicands

√(−x) for x > 0 is not a real number. The calculator reports the imaginary form (e.g. √(−9) = 3i) so the algebra stays consistent without silently dropping the sign.

Decimals and fractions

Square roots of non-integer values still simplify — √0.25 = 0.5, √(9/16) = 3/4. The calculator returns the precise decimal for any real input.

Positive vs principal root

Every positive number technically has two square roots — a positive and a negative one. By convention √x denotes the principal (positive) root; the equation x² = 4 has solutions x = ±2.

Core Formulas

Definition

√x = y ⇔ y² = x (x ≥ 0)

The defining identity: square the root and the radicand returns.

Product rule

√(a·b) = √a · √b (a, b ≥ 0)

Lets you factor radicals into smaller pieces and simplify.

Quotient rule

√(a/b) = √a / √b (a ≥ 0, b > 0)

Distributes the radical over a fraction so each part can simplify on its own.

Exponent form

√x = x^(1/2)

Square root as a fractional exponent — interchangeable with the radical.

Imaginary unit

√(−1) = i

Defines the imaginary unit; lets √(−x) be written as √x · i for x > 0.

Simplified form

√(k²·m) = k·√m

Pull out the largest perfect-square factor of the radicand for the simplified radical.

Common Mistakes

Treating √(a + b) as √a + √b

The radical does not distribute over a sum. √(9 + 16) = √25 = 5, not √9 + √16 = 7. Only the product rule, not the sum rule, holds.

Dropping the imaginary sign

Returning a positive number for √(−4) is wrong — the answer is 2i, not 2. Calculators that silently flip the sign give misleading results in algebra and physics.

Confusing principal and full solutions

√x denotes the positive root by convention. When solving x² = 9 the answer is x = ±3, but √9 by itself is only 3.

Forgetting to verify

Type the wrong number and you might still get a plausible-looking decimal. Always square the answer to confirm — the calculator shows the verification line for this reason.

Built for students, engineers, scientists, and anyone working with radical expressions — from a quick √144 check to a full nth-root domain analysis. Every result is verified by squaring (or cubing, or raising to the n-th power) the answer back so you can confirm the math without leaving the page.

Square Root Calculator FAQs

The square root of a non-negative number x, written √x, is the value that gives x when multiplied by itself. For example, √144 = 12 because 12 × 12 = 144, and √50 = 5√2 because 50 = 25 × 2 and √25 = 5. The square root is the inverse of squaring (raising to the second power), and by convention √x denotes the principal (positive) root. Negative radicands yield imaginary numbers — √(−4) = 2i, where i = √(−1).

Factor the radicand into the largest perfect-square divisor times a square-free remainder, then pull the square root of the perfect square out front. For √50, write 50 = 25 × 2, then √50 = √25 · √2 = 5√2. For √72, write 72 = 36 × 2, so √72 = 6√2. For √200, write 200 = 100 × 2, so √200 = 10√2. The calculator does this automatically using a prime-factor scan.

A perfect square is an integer that equals an integer squared. The first twelve perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 — corresponding to 1² through 12². Square roots of perfect squares are always exact integers. Recognising the small perfect squares lets you simplify radicals quickly: any radicand that has one as a factor will simplify cleanly.

Yes. The square root of any non-negative real number is itself a real number; if the radicand isn't a perfect square the answer is irrational and only has a decimal approximation. √2 ≈ 1.41421356, √3 ≈ 1.73205081, √5 ≈ 2.23606798. Decimals also apply to non-integer radicands: √0.25 = 0.5, √0.04 = 0.2, √2.5 ≈ 1.58113883. The calculator returns ~15 significant digits for any decimal result.

Not in the real numbers. The square of any real number — positive or negative — is non-negative, so no real number squares to a negative value. The square root of a negative number is therefore imaginary: √(−1) = i by definition, and √(−x) for x > 0 is written as √x · i. For example, √(−9) = 3i and √(−50) = 5i√2. The calculator returns the imaginary form whenever you enter a negative radicand.

The cube root of x, written ∛x, is the number that gives x when raised to the third power. ∛27 = 3 because 3³ = 27, and ∛125 = 5 because 5³ = 125. Unlike square roots, cube roots are defined for every real number — including negatives. ∛(−8) = −2 because (−2)³ = −8. Switch to the Cube tab above to compute cube roots with full step-by-step working.

The n-th root of x, written ⁿ√x or equivalently x^(1/n), is the number that gives x when raised to the n-th power. n = 2 is the square root, n = 3 is the cube root, n = 4 is the fourth root, and so on. The General tab solves any positive integer (or fractional) index. Domain rule: when n is even the radicand must be ≥ 0 to stay in the real numbers; when n is odd any real radicand is allowed.

Yes — every root is a fractional exponent. ⁿ√x = x^(1/n), so √x = x^(1/2), ∛x = x^(1/3), and ⁴√x = x^(1/4). Mixed fractional exponents like x^(2/3) mean 'cube root of x, then squared' — equivalent to (∛x)². Switching radical notation to exponent notation often simplifies algebra because the integer-exponent rules (xᵃ · xᵇ = x^(a+b), (xᵃ)ᵇ = x^(a·b)) apply directly.

Every positive number has two square roots — a positive and a negative one — because both (+k)² and (−k)² equal k². By convention √x denotes only the positive (principal) root. When solving an equation like x² = 9, the solutions are x = ±3, but √9 by itself is only 3. The same convention applies to higher even roots — ⁴√16 means the positive value 2, not ±2.

The calculator uses double-precision floating-point arithmetic (about 15–16 significant digits), which exceeds practical engineering and scientific needs by a wide margin. Perfect squares, cubes, and n-th powers are detected exactly so the calculator returns exact integers when one applies. Every result includes a verification line (square the answer, cube it, or raise it to the n-th power) so you can confirm the math by hand at a glance.

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Pair the root calculator with other math tools for algebra, geometry, statistics, and coordinate analysis.