Square Root Calculator

Calculate square roots, cube roots, and nth roots. Simplify radicals and identify perfect squares.

Something wrong or missing? Tell us — we read every report and fix issues fast.

How to Calculate Roots

A root is the inverse of an exponent. The square root of a number asks: what value multiplied by itself gives this number? This calculator shows both exact and decimal results for square roots, cube roots, and nth roots.

Square Root (√)

Find what number × itself = your number.

√4 = 2 (because 2 × 2 = 4)
√9 = 3 (because 3 × 3 = 9)
√2 ≈ 1.41421 (irrational — decimal never terminates or repeats)

Cube Root (∛)

Find what number × itself × itself = your number.

∛8 = 2 (because 2 × 2 × 2 = 8)
∛27 = 3
∛-8 = -2 (cube roots can be negative, unlike square roots)

Perfect Squares (1-400)

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400

Recognizing perfect squares makes mental math and radical simplification much faster.

How to Simplify Radicals

To simplify a square root, factor out the largest perfect square. The product rule states: √(a × b) = √a × √b.

Example: √72

Find the largest perfect square factor: 72 = 36 × 2
√72 = √(36 × 2) = √36 × √2
= 6√2 ≈ 8.485

More Examples

√50 = √(25 × 2) = 5√2
√200 = √(100 × 2) = 10√2
√48 = √(16 × 3) = 4√3
√75 = √(25 × 3) = 5√3
√98 = √(49 × 2) = 7√2

Square Roots in Real Life

Distance formula: The distance between two points uses √((x₂-x₁)² + (y₂-y₁)²)
Pythagorean theorem: Find the hypotenuse with c = √(a² + b²). A 3×4 rectangle has a diagonal of √(9+16) = √25 = 5
Standard deviation: Statistical spread is calculated as the square root of variance
Screen size: TV diagonal = √(width² + height²). A 16:9 TV that's 44" wide × 24.75" tall has a diagonal of √(1936 + 612.56) = √2548.56 ≈ 50.5"

Properties of Square Roots

√(a × b) = √a × √b — product rule
√(a ÷ b) = √a ÷ √b — quotient rule
(√a)² = a — squaring undoes the root
Negative numbers have no real square root. √(-1) = i (imaginary unit)
Every positive number has two square roots: positive and negative. √25 = ±5, though the symbol √ conventionally means the positive root

Related Calculators

Percentage Calculator — 7 Modes, Step-by-Step Work Calculate percentages 7 ways: find percent, find whole, percent change, percent difference, add/subtract percent, and reverse percent. Fraction Calculator — Step-by-Step Solver Add, subtract, multiply, divide, compare, and solve fractions with full step-by-step work. Chain up to 9 fractions, mixed numbers, simplify, and decimal conversions. Equivalent Fractions Calculator — First 100 Results Find equivalent fractions for any proper, improper, or mixed fraction. Shows the first 100 equivalent fractions instantly. Average Calculator — Mean, Median, Mode, Range, SD Calculate mean (sum divided by count), median, mode, range, and standard deviation with step-by-step work. Includes weighted average mode and a full statistics measures reference. Ratio Calculator — Simplify, Solve, Scale, Compare Simplify ratios, solve proportions, scale, split, compare, and generate equivalent ratio tables — all in one tool. Exponent Calculator — Powers, Roots, Logs, e^x Calculate powers, roots, exponentials, and logarithms. Solve xⁿ, ⁿ√x, eˣ, and log expressions with full work shown.

Frequently Asked Questions

What is the square root of 50?

The square root of 50 is about 7.07. It lies between the perfect squares 49 and 64, so its root is between 7 and 8 and much closer to 7. A quick estimate is 7 + (50 - 49) / (64 - 49) ≈ 7.07. In simplest radical form, √50 = √(25 × 2) = 5√2.

How do you simplify the square root of 72?

√72 simplifies to 6√2. Find the largest perfect square that divides the number, here 36, and take its root outside: √72 = √(36 × 2) = √36 × √2 = 6√2. Perfect squares to look for include 4, 9, 16, 25, 36, 49, 64, 81 and 100. Stopping at a smaller factor such as 4 leaves the radical only partly simplified.

Can you take the square root of a negative number?

Not in the real numbers. No real number multiplied by itself gives a negative result, so the square root of a negative number is imaginary. It is written using i = √(-1): for example √(-16) = 4i. The square root of 0 is defined and equals 0, because 0 × 0 = 0.

Does x² = 9 have one solution or two?

It has two solutions, x = 3 and x = -3, because both 3² and (-3)² equal 9. The √ symbol on its own means the principal square root, which is never negative, so √9 = 3. When solving an equation by taking a square root, write ±√ to keep both solutions; dropping the negative one is a common mistake.