Square Root Calculator
The decimal, and the simplified radical form a phone will not give you.
Updated
Negatives are allowed — the root is imaginary.
Square root
8.485281
Not a perfect square, so the decimal never terminates.
- Simplified radical form
- 6√2
- Perfect square
- No
- Cube root
- 4.160168
- Rounded to 4 places
- 8.4853
72 splits into 36 × 2, and 36 is a perfect square. Pulling it out leaves 6√2, which is exact — unlike the decimal, which is an irrational number that no finite number of digits can write down.
In short
What is the square root of 72 in simplified radical form?
6√2. Seventy-two factors into 36 × 2, and 36 is a perfect square, so its root comes outside: √72 = √36 × √2 = 6√2. As a decimal that is 8.4853, but the radical form is exact and the decimal never terminates.
The decimal is an approximation. The radical is the number itself.
How to use the square root calculator
The square root of a number is the value that, multiplied by itself, gives that number back. Twelve times twelve is 144, so the square root of 144 is 12. Most numbers are not perfect squares, and their roots are irrational — decimals that continue forever without repeating, which no finite string of digits can write down exactly.
That is why simplified radical form matters. Instead of rounding √72 to 8.4853 and losing precision, you can rewrite it as 6√2, which is exactly equal to it. The method is to find the largest perfect square that divides the number, take its root outside the radical, and leave the rest inside. Seventy-two is 36 × 2, and 36 is 6².
6√2
The square root of 72
exact, unlike 8.4853
1.41421356
The square root of 2
and it never terminates
5i
The square root of −25
imaginary, not an error
Negative numbers have no real square root, because multiplying any real number by itself gives a positive result. Mathematics answers this by defining i as the square root of −1, which makes the square root of −25 equal to 5i. That is a genuine answer rather than a failure, and it is what makes the quadratic formula work when its discriminant goes negative.
Cube roots behave differently and it is worth knowing why. An odd power preserves sign, so −2 cubed is −8 and the cube root of −8 is a perfectly real −2. Only even roots have the problem, and only for negative inputs. That asymmetry catches people who assume all roots of negatives are imaginary.
- No perfect square factor left
- inside the radical
- No fraction
- under the radical sign
- No radical
- in a denominator
- For an nth root
- no perfect nth power inside
- It is a convention
- for presentation, not a law of arithmetic
The last row is worth stating plainly: a radical in a denominator is not mathematically invalid, and rationalising it is a school-algebra tidiness rule rather than a correctness requirement.
Where square roots come up most
The quadratic formula has a square root at its heart, and whether the number under it is a perfect square decides whether the roots are tidy.
Open the quadratic calculator →Square roots in simplified radical form beside their decimal values, showing which numbers reduce and which are already as simple as they get.
| Number | Simplified | Decimal |
|---|---|---|
| √8 | 2√2 | 2.8284 |
| √12 | 2√3 | 3.4641 |
| √18 | 3√2 | 4.2426 |
| √20 | 2√5 | 4.4721 |
| √32 | 4√2 | 5.6569 |
| √50 | 5√2 | 7.0711 |
| √72 | 6√2 | 8.4853 |
| √98 | 7√2 | 9.8995 |
| √144 | 12 | 12 |
| √200 | 10√2 | 14.1421 |
| √13 | √13 | 3.6056 |
Finding the largest perfect square
The method is to work downwards. Start from the largest whole number whose square could divide your value and test each one until something fits. For 72, test 8² = 64 (no), 7² = 49 (no), 6² = 36 (yes, 72 ÷ 36 = 2). Stop there: the first one that divides is the largest, and the answer is 6√2.
- Largest square factor
- 72 = 36 × 2 → 6√2
- Prime factors instead
- 72 = 2 × 2 × 2 × 3 × 3
- Pair them off
- (2×2), (3×3), and a spare 2
- Each pair comes out as
- 2 and 3
- Which multiply to
- 6, leaving √2 inside
The prime factorisation route is slower to write and impossible to get wrong: every pair of identical primes contributes one of them to the outside, and anything unpaired stays under the radical. It also generalises directly to cube roots, where you look for triples instead of pairs.
The check is always available. Whatever comes out, squaring the outside coefficient and multiplying by whatever remains inside must return the original number: 6² × 2 = 72. If it does not, either the factor was not a perfect square or the division was wrong.
Why most square roots never end
A square root is rational only when the number is a perfect square. Everything else produces an irrational number — a decimal that continues forever without ever settling into a repeating pattern. The square root of two is the standard example, and its irrationality has a proof short enough to fit in a paragraph.
Do
- Keep the radical form when the answer feeds into further work
- Use √(a²) = |a| rather than dropping the absolute value
- Pair off prime factors to simplify without guessing
- Check by squaring the outside and multiplying back
Don't
- Assume √(a + b) equals √a + √b — it almost never does
- Treat a rounded decimal as the exact value in a chain of calculations
- Expect the cube root of a negative to be imaginary
- Call a radical in a denominator mathematically wrong
That first "don't" is the most consequential. The square root of 9 plus 16 is the square root of 25, which is 5 — not 3 plus 4. Roots do not distribute over addition, only over multiplication and division, and assuming otherwise produces answers that are wrong by a wide and unpredictable margin.
The formula, worked line by line
One definition, one rule for pulling factors out, and one identity that is more often written wrongly than any other on this page.
“A root distributes over multiplication and division. It does not distribute over addition, and no amount of wanting it to will change that.”
Everything about simplifying radicals follows from the first half of that sentence. Splitting 72 into 36 times 2 works precisely because the root of a product is the product of the roots.
√(a × b) = √a × √b
√(a ÷ b) = √a ÷ √b
√(a²) = |a|, not a
√(−n) = i√n
√(a + b) ≠ √a + √b- Suppose it were
- a fraction p/q in lowest terms
- Then
- p² = 2q², so p² is even
- So p is even
- write it as 2k
- Substituting
- 4k² = 2q², so q² = 2k²
- So q is even too
- but p/q was in lowest terms
- Contradiction
- no such fraction exists
The argument is a proof by contradiction and it needs nothing beyond the fact that a square is even only when its root is. It is associated with the Pythagorean school, though the familiar story about Hippasus being drowned for revealing it is a much later legend rather than recorded history.
Every positive number has two square roots
Both 5 and −5 square to 25, so both are square roots of 25. The radical symbol √ denotes only the non-negative one, called the principal square root, which is why √25 is 5 rather than "5 or −5". When solving x² = 25 you must write ±5 explicitly, because there the equation asks for both.
Roots that come out whole
- 1, 4, 9, 16, 25
- roots 1 to 5
- 36, 49, 64, 81, 100
- roots 6 to 10
- 121, 144, 169, 196, 225
- roots 11 to 15
- 256, 289, 324, 361, 400
- roots 16 to 20
- 625, 900, 1024
- 25, 30 and 32
Knowing these on sight is what makes simplifying fast: you are looking for the largest one that divides your number, and the search is much quicker when the candidates are already familiar.
It also makes the perfect-square test instant. If a number ends in 2, 3, 7 or 8 it cannot be a perfect square at all, because no square ends in those digits — a quick filter before any dividing starts.
Questions people ask
Sources
Where the constants and formulas on this page come from. Each line names the figure it backs.
The radical symbol denotes the principal, non-negative square root, and every positive real number has two square roots.
Square Root — Wolfram MathWorld
The square root of two is irrational, provable by contradiction from the assumption that it is a ratio of integers in lowest terms.
Pythagoras's Constant — Wolfram MathWorld
Simplified radical form requires no perfect nth power under an nth root, no fraction under the radical, and no radical in a denominator.
Simplify Radical Expressions — Intermediate Algebra — OpenStax
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