Arithmetic
Square Root Calculator
Find the square root of any number — its decimal value, its simplified radical form, and whether it's exact or irrational — with every step shown. Switch to cube and nth roots, and scan the lookup charts.
Find a root
Root type
√72 =
8.485281
irrational · approximate
Irrational — no exact whole-number root6√2
Between 8 and 9 (8² = 64, 9² = 81)
8.485281²
72
n²
5184
± roots
±8.485281
Try an example
No sign-up · Free
The basics
What is a square root?
A square root reverses squaring. The square root of a number is the value that, multiplied by itself, returns that number — so it answers “what number times itself equals this?” Some answers are clean whole numbers; most are not, and we have two ways of writing those.
√n = x ⇔ x² = n
The definition
The radical sign √ asks for the number that squares back to n. By convention it returns the non-negative principal root.
√25 = 5 because 5² = 25
√(a · b) = √a · √b
Exact vs approximate
When a perfect square hides inside, factor it out for an exact radical form; otherwise the root is irrational and you give a rounded decimal.
√72 = 6√2 ≈ 8.485281
Method
How to simplify a square root by hand
Three steps turn an ugly radical into its simplest exact form. We'll simplify √72.
- 1
Find the largest perfect-square factor
List the perfect squares (4, 9, 16, 25, 36…) and find the biggest one that divides n evenly.
72 = 36 × 2, and 36 is a perfect square - 2
Split the radical across that factor
Use √(a · b) = √a · √b to break the root into a perfect-square part and a leftover part.
√72 = √36 × √2 - 3
Take the root you can, leave the rest
Pull the whole-number root outside the radical. Whatever remains underneath is the simplified form.
√36 × √2 = 6√2 - 4
Approximate if you need a decimal
The leftover radical is irrational, so round it to as many places as the task needs.
6√2 ≈ 8.485281
Worked examples
See it across the four cases
Every square root falls into one of four situations — exact, simplifiable, irrational, or imaginary. Switch tabs to see each worked out.
Given Find √144
Look for an integer that squares to 144: 12 × 12 = 144
So the root is exact.
√144 = 12
Quick chart
What is the square root of X?
A scannable reference for the searches people actually type — perfect squares to recognise, and the decimal value of every root from √1 to √100.
1²
1
2²
4
3²
9
4²
16
5²
25
6²
36
7²
49
8²
64
9²
81
10²
100
11²
121
12²
144
13²
169
14²
196
15²
225
16²
256
17²
289
18²
324
19²
361
20²
400
21²
441
22²
484
23²
529
24²
576
25²
625
26²
676
27²
729
28²
784
29²
841
30²
900
Memorising squares 1–30 lets you recognise and simplify most radicals on sight.
Estimation
Approximating roots without a calculator
Before electronics, people squeezed square roots out by hand. Two techniques still make great mental-math shortcuts.
lo² ≤ n ≤ hi²
Bracket and interpolate
Pin n between the two nearest perfect squares, then judge how far along it sits. The closer n is to the lower square, the closer the root is to the lower whole number.
√50: between 7²=49 and 8²=64 → ≈ 7.07
xₖ₊₁ = (xₖ + n / xₖ) / 2
The Babylonian method
Guess, divide n by the guess, average the two, repeat. Each pass roughly doubles the correct digits — it converges astonishingly fast.
√10: 3 → 3.166667 → 3.162281 → 3.162278 → 3.162278 (→ 3.162278)
Where it shows up
Why square roots matter
The operation that undoes squaring sits at the heart of geometry, statistics, and physics.
c = √(a² + b²)
Distance & geometry
The Pythagorean theorem and the distance formula both end in a square root — it converts squared lengths back into a real distance.
σ = √(variance)
Statistics
Standard deviation is the square root of the variance, pulling a spread measured in squared units back to the original scale.
T ∝ √L
Physics
A pendulum's period grows with the square root of its length, and many natural laws carry the same square-root relationship.
Worth knowing
Rules and pitfalls worth remembering
A handful of identities make roots easier to manipulate — and one famous trap to avoid.
√(a · b) = √a · √b
Product rule
The root of a product is the product of the roots. This is the rule that powers radical simplification.
√(a / b) = √a / √b
Quotient rule
The root distributes over division too, letting you simplify fractions inside a radical.
√(a + b) ≠ √a + √b
The classic mistake
Roots do not distribute over addition. This is the single most common error students make with radicals.
√(n²) = |n|
Mind the sign
The principal square root of a square is the absolute value, not the original number — it is always non-negative.
x² = n ⇒ x = ±√n
Two roots when solving
Solving an equation keeps both signs, even though the √ symbol alone returns just the positive root.
√(aᵐ) = a^(m/2)
Roots are fractional powers
A square root is the same as raising to the one-half power, which is why root and exponent rules line up.
Variants
Beyond square: cube and nth roots
The same idea generalises to any power. Switch the tool's mode to compute these, and note one key difference about negatives.
∛n = x ⇔ x³ = n
Cube root
Undoes cubing. Unlike square roots, cube roots accept negatives — a negative cubed stays negative.
∛64 = 4, ∛−27 = −3
ⁿ√x = x^(1/n)
The nth root
The general root reverses raising to the nth power, and equals a fractional exponent.
⁴√81 = 3, ⁵√32 = 2
even vs odd n
When negatives work
Even roots of negatives are imaginary; odd roots of negatives are real. The parity of n decides.
√−4 = 2i, ∛−8 = −2
FAQ
Common questions
The square root of a number is the value that, multiplied by itself, gives that number. For perfect squares you can read it off exactly (√25 = 5, because 5 × 5 = 25). For other numbers the root is irrational, so you either approximate it as a decimal — √2 ≈ 1.414214 — or factor out the largest perfect square to write it in simplified radical form.
Find the largest perfect-square factor of the number, split the radical across that factor, and pull its root outside. To simplify √72, the largest perfect square dividing 72 is 36, so √72 = √(36 × 2) = √36 × √2 = 6√2. If no perfect square other than 1 divides the number, it is already in simplest form.
Not within the real numbers. No real number squared gives a negative result, because a negative times a negative is positive. The square root of a negative number is imaginary — √−1 is written as i — and √−9 = 3i. The calculator flags this instead of returning a real value.
A perfect square is an integer that is the square of another integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. Their square roots are whole numbers (√49 = 7), so they have exact, non-decimal roots. Knowing the squares up to 30² helps you simplify radicals quickly.
No. √2 is irrational — it cannot be written as a fraction of two integers, and its decimal expansion (1.41421356…) never repeats or terminates. This was one of the first numbers proven irrational. Most square roots of non-perfect-squares are irrational in the same way.
Every positive number has two square roots: a positive (principal) root and its negative. Both 5 and −5 square to 25, so the solutions of x² = 25 are x = ±5. The radical symbol √ by convention returns only the non-negative principal root, which is why √25 = 5, not ±5.
Find the two perfect squares the number sits between and interpolate. √50 lies between √49 = 7 and √64 = 8, and since 50 is just past 49, the root is just over 7 (≈ 7.07). For more precision, use the Babylonian method: guess, divide, average, and repeat — it doubles the accurate digits each step.
A square root undoes squaring (raising to the power 2), so √n is the number that, squared, gives n. A cube root undoes cubing (power 3): ∛n is the number that, cubed, gives n. Cube roots, unlike square roots, are defined for negative numbers — ∛−8 = −2 — because a negative cubed stays negative.
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Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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