How square roots work
The square root of a number is the value that multiplies by itself to produce that number:
If √n = r, then r × r = n
So √25 = 5 because 5 × 5 = 25. When a number isn't a perfect square, its root is an irrational decimal that never ends or repeats — √72 ≈ 8.485 — which is why the exact radical form is often more useful.
Simplifying a radical
To simplify, find the largest perfect square that divides your number, then take its root out front. For 72, the largest perfect-square factor is 36:
√72 = √(36 × 2) = √36 × √2 = 6√2
The calculator does this for any whole number. If the result is a single whole number with no radical, your number is a perfect square.
Common perfect squares
| Number | Square root |
|---|---|
| 1, 4, 9, 16, 25 | 1, 2, 3, 4, 5 |
| 36, 49, 64, 81, 100 | 6, 7, 8, 9, 10 |
| 121, 144, 169, 196, 225 | 11, 12, 13, 14, 15 |
Frequently asked questions
How do you find the square root of a number?
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5 because 5 × 5 = 25. For numbers that are not perfect squares, the square root is an unending decimal, which this calculator gives to several places — plus the exact simplified radical form.
What is a simplified radical form?
It rewrites a square root by pulling out the largest perfect-square factor. For example, √72 = √(36 × 2) = 6√2. This exact form is preferred in algebra because it keeps the value precise instead of rounding to a decimal. The calculator shows it automatically for whole numbers.
What is a perfect square?
A perfect square is a number whose square root is a whole number — 1, 4, 9, 16, 25, 36, and so on. If the simplified radical comes out as a single whole number with no radical sign, your number is a perfect square.
Can you take the square root of a negative number?
Not within the real numbers — no real number squared gives a negative result. The square root of a negative number is an imaginary number (for example, √−9 = 3i). The cube root of a negative number, however, is a real number, since a negative times a negative times a negative is negative.
Note: The square root of a negative number is imaginary and is shown as such. Decimal results are rounded for display.