Square Root Calculator
Enter any number to find its square root as a decimal and in simplest radical form — for example √72 = 6√2 — with every step shown. Detects perfect squares and gives the cube root too.
- 01Find the square root of any number as an exact decimal instantly.
- 02Simplify square roots into simplest radical form, like √72 = 6√2.
- 03Detect perfect squares and see the simplified integer result.
- 04Get the cube root and the original number alongside each result.
- 05100% free and private — every calculation runs in your browser.
Square Root Calculator
Try an example number
Square Root (√N)
8.485281
The principal square root as a decimal
Simplest Radical Form
62
Exact form with the largest perfect square factored out
Is Perfect Square
No
Cube Root (∛N)
4.160168
Original Number
72
Step-by-step calculation
- 01Factor out the largest perfect square: 72 = 36 × 2.
- 02Split the radical: √{{n}} = √36 × √2 = 6√2.
- 03Simplest radical form: √N = 6√2 ≈ 8.485281.
Why Use This Square Root Calculator
Decimal and Exact Radical Form
Get the square root of a number both ways: a precise decimal value and the exact answer in simplest radical form. For example, the calculator shows √72 as 6√2 and as 8.485281, so you can use whichever form your homework, exam, or project requires.
Simplify Radicals Automatically
This square root simplifier factors out the largest perfect square so every radical is reduced to its simplest form with a square-free number under the root. Enter √200 and it returns 10√2; enter √45 and it returns 3√5 — no manual factoring required.
Perfect Square Detection
Instantly see whether a number is a perfect square. If it is, the radical disappears and you get a clean whole-number result — √144 = 12, √169 = 13 — making it easy to spot perfect squares while practising or checking work.
Step-by-Step Working
See exactly how each square root is simplified: identifying the largest perfect square factor, splitting the radical, and pulling the coefficient out front. The step-by-step explanation makes this a great tool for learning how to simplify a square root by hand.
Cube Roots and More
Beyond the square root, the calculator also reports the cube root of the number and echoes the original value, giving you a quick, complete summary of the number's roots in one place — handy for comparing radicals and roots.
Instant, Free, and Private
Everything runs entirely in your browser with no server round-trips, no signup, and no limits. Your numbers never leave your device, results appear the moment you type, and you can simplify as many radicals as you need for free.
What Is a Square Root?
The square root of a number N is the value that, when multiplied by itself, equals N. It is written with the radical symbol √, so √25 = 5 because 5 × 5 = 25. Every positive number has two square roots — one positive and one negative — but the radical symbol refers to the principal (non-negative) square root. Square roots are fundamental in algebra, geometry, the Pythagorean theorem, statistics, and physics.
Whether you are a student simplifying radicals for algebra, a teacher preparing examples, or anyone who needs the square root of a number fast, this square root calculator gives instant decimal values, exact simplest radical form, and clear step-by-step explanations.
- Perfect Squares vs Irrational Roots
- A perfect square is a number whose square root is a whole number, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. When N is not a perfect square, its square root is an irrational number with a non-repeating decimal — for example √2 ≈ 1.41421356 — which is why writing it in exact radical form is often preferred.
- Simplest Radical Form
- Simplifying a square root means rewriting it so the number under the radical (the radicand) has no perfect-square factors other than 1. To do this, find the largest perfect square that divides N, take its root out front as a coefficient, and leave the rest under the radical. For example √72 = √(36 × 2) = 6√2.
- How to Simplify a Square Root
- Factor the radicand into a perfect square times a remaining factor, split the radical using √(a × b) = √a × √b, then evaluate the perfect-square part. The square-free number stays under the radical. This is the standard method this radical calculator automates and shows step by step.
- Square Roots of Negative Numbers
- The square root of a negative number is not a real number; it is imaginary. The calculator expresses it as a multiple of i, where i = √-1. For example √-9 = 3i, because (3i)² = 9i² = -9. Imaginary and complex numbers extend the real number line and are essential in advanced mathematics and engineering.
How to Use the Square Root Calculator
- 01
Enter a number
Type or paste a single number into the input box — for example 72. Decimals such as 0.25 and negative numbers such as -16 are supported, so you can find the square root of almost any value.
- 02
Click Calculate
Press the Calculate button (or hit Enter). The tool computes the square root, simplifies the radical, checks whether the number is a perfect square, and finds the cube root instantly in your browser.
- 03
Read the results
The headline cards show the decimal square root and the simplest radical form (for example 6√2). Below them you can see whether the number is a perfect square, its cube root, and the original number you entered.
- 04
Review the steps
Open the step-by-step section to see how the radical was simplified: the largest perfect square factor, how the radical splits, and how the coefficient comes out front. Great for learning and for checking your own work.
Tips for Simplifying Square Roots
Memorise the Perfect Squares
Knowing the perfect squares up to 144 or higher (1, 4, 9, 16, 25, …, 144) makes simplifying radicals much faster, because you can immediately spot the largest perfect square factor inside any radicand.
Factor Out the Largest Square
Always factor out the largest perfect square in one step rather than several. √72 simplifies directly to 6√2 using 36, instead of 2√18 using 4 — which still needs more simplifying. Pulling the biggest square first saves work.
Leave Square-Free Radicands
A square root is in simplest form only when the number under the radical has no perfect-square factors except 1. If you can still factor out a square, the radical is not fully simplified yet — keep going until the radicand is square-free.
Keep Exact Form When Possible
For exact answers, prefer the radical form (like 6√2) over a rounded decimal. Decimals introduce rounding error, while the simplified radical is exact — important in algebra, geometry, and trigonometry.
Watch for Negative Inputs
The square root of a negative number is imaginary, not undefined. Express it using i (for example √-25 = 5i). Don't confuse this with the cube root, which is defined for negative numbers and stays real.
Check With the Decimal
After simplifying, verify your radical by comparing its decimal value to the calculator's. If 6√2 and 8.485281 match, your simplification is correct — a quick sanity check that catches factoring mistakes.
Square Root Formulas and Definitions
Definition of a square root
The square root of N is the number x such that x² = N. The principal square root, written √N, is the non-negative value. For positive N there are two square roots, +√N and -√N, but √N by convention denotes the positive one.
What simplifying a radical means
- Rewrite √N as a coefficient times a smaller radical, e.g. √72 = 6√2.
- Factor out the largest perfect square so the radicand is square-free.
- Keep exact values instead of rounding to a decimal.
- A perfect square simplifies to a whole number with no radical left.
Square roots vs cube roots
A square root undoes squaring (x²), while a cube root undoes cubing (x³). Square roots of negative numbers are imaginary, but cube roots of negative numbers are real — for example ∛-27 = -3.
Key Square Root Formulas
Definition
√N = x where x ≥ 0 and x² = N
Example: √25 = 5 because 5² = 25.
Product rule for radicals
√(a × b) = √a × √b
Used to split off a perfect square, e.g. √72 = √36 × √2.
Simplest radical form
√N = c√r, where c² is the largest perfect square factor of N and r is square-free
Example: √72 = 6√2, since 72 = 36 × 2.
Perfect square
N is a perfect square if √N is an integer
Example: 144 is a perfect square because √144 = 12.
Imaginary square root
√(-N) = √N · i, where i = √-1
Example: √-9 = 3i.
Cube root
∛N = x where x³ = N
Example: ∛64 = 4, and ∛-27 = -3.
Square Root Calculator FAQ
Q01How do I find the square root of a number?
The square root of a number N is the value that multiplies by itself to give N. For perfect squares the answer is a whole number — √81 = 9. For other numbers it is an irrational decimal, such as √10 ≈ 3.162278. This square root calculator gives both the decimal value and, where possible, the exact simplest radical form, plus step-by-step working.
Q02How do I simplify a square root?
To simplify a square root, find the largest perfect square that divides the number under the radical, take its root out front as a coefficient, and leave the remaining square-free factor under the radical. For example √72 = √(36 × 2) = 6√2. The calculator does this automatically and shows each step.
Q03What is simplest radical form?
Simplest radical form means the number under the radical (the radicand) has no perfect-square factors other than 1, and there are no radicals in any denominator. For example √50 in simplest radical form is 5√2, and √45 is 3√5. If a number is a perfect square, its simplest form is just a whole number.
Q04What is a perfect square?
A perfect square is a number whose square root is a whole number, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. When you take the square root of a perfect square, the radical disappears and you get an integer — for example √144 = 12. The calculator flags whether your number is a perfect square.
Q05Can this calculator simplify radicals like √72?
Yes. Enter 72 and the calculator returns the simplest radical form 6√2 along with the decimal 8.485281. It works as a radical calculator and square root simplifier for any non-negative whole number, factoring out the largest perfect square automatically.
Q06What is the square root of a negative number?
The square root of a negative number is imaginary, written as a multiple of i, where i = √-1. For example √-9 = 3i because (3i)² = -9. This calculator detects negative inputs and shows the result in the form Ni, so you can work with imaginary numbers correctly.
Q07Does it work for decimals?
Yes. You can enter decimal numbers such as 0.25 or 2.5, and the calculator returns the decimal square root — for example √0.25 = 0.5. Simplest radical form is only shown for whole numbers, since radical simplification applies to integers; for decimals the calculator shows the decimal value.
Q08Is this square root calculator free?
Yes, it is completely free with no limits, no signup, and no premium tier. Calculate square roots, simplify radicals, detect perfect squares, and find cube roots for as many numbers as you like, entirely in your browser.