The rule
√n = the number that gives n when squared
Why it works
A square root asks which number, multiplied by itself, gives the one you started with. When that answer is a whole number the root is exact; otherwise it is irrational, and the radical form is the only way to write it precisely.
How to do it by hand
- Break the number into prime factors
- Any factor appearing twice comes out of the radical once
- What is left stays inside
- Multiply out what came out for the simplified form
What is worth knowing
Simplifying radicals matters more than it looks. √72 = 6√2 is exact and 8.485 is not, and in any calculation that continues, carrying the exact form avoids compounding rounding errors. The Greeks discovered that √2 cannot be written as a fraction, and the proof — assume it can, show the fraction can always be reduced further, contradiction — is one of the oldest arguments in mathematics still taught unchanged. Negative numbers have no real square root, because no real number squares to a negative; that gap is exactly where the imaginary unit comes from.