Square Root Calculator
Most square roots are irrational, so the exact form matters more than the decimal. This calculator finds the largest square factor and shows where it came from.
How it works
How the Square Root Calculator works
the part a result on its own leaves out
What simplifying a root means
√72 and 6√2 are the same number. The second is simpler because the number left under the root has no square factor, which makes it easier to combine with other roots and impossible to simplify further.
How the square factor is found
The calculator divides out each prime that appears an even number of times. That guarantees what remains under the root is square-free — not just smaller, but as small as it can be.
Common mistakes
Common mistakes with square roots
the errors that actually show up, not every possible slip
Splitting a sum
√(9 + 16) is 5, not 3 + 4. Roots distribute over multiplication and division, never over addition or subtraction.
Stopping at the first square factor
√72 = 2√18 is true but not finished. 18 still contains 9, so the simplest form is 6√2.
Rounding too early
Turning √2 into 1.41 at the start compounds the error through every later step. Keep the root exact until the final line.
Questions
Asked often enough to answer here
answers that change how you work
Why can't I take the square root of a negative number?
No real number squared gives a negative result. It becomes possible with imaginary numbers, where i is defined as √−1.
Is √n always irrational?
Only when n is not a perfect square. √49 is exactly 7; √50 cannot be written as any fraction at all.
Does every root simplify?
No. If the number has no square factor above 1 — 2, 3, 5, 6, 7, 10 and so on — it is already in simplest form, and the calculator says so.
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