Square roots and perfect squares
The square root of a number n is the non-negative number which, multiplied by itself, gives n. The square root of 144 is 12, because 12 × 12 = 144. The square root of 0 is 0. A negative number has no real square root: no real number squares to −4. The calculator refuses those instead of inventing an imaginary result. Cube roots are different. The cube root of −8 is −2, because −2 × −2 × −2 = −8, and that option is in the menu.
Numbers that are not perfect squares still have a decimal square root. The square root of 2 is about 1.414214. The square root of 50 is about 7.0711. You can check a result by squaring it. Squaring a rounded root will land very near the original number, and the last digits may differ because the display stopped at eight decimal places.
Worked example: simplifying √72
An integer under a radical can often be rewritten as a smaller radical times a whole number. Factor out the largest perfect square.
- 72 = 36 × 2, and 36 is 6 × 6.
- So √72 = √(36 × 2) = 6√2.
- The decimal is 6 × 1.414214 = 8.485281.
- Check: 8.485281 squared is 72, within display rounding. The simplified form is 6√2.
Estimating before you trust the screen
A square root sits between the roots of the perfect squares around it. √50 is between √49 = 7 and √64 = 8, and closer to 7 because 50 is just above 49. The calculator’s value, about 7.071, matches that guess. √10 is between 3 and 4, closer to 3, and the value is about 3.162. If a result falls outside the neighboring perfect squares, it is wrong.
Square roots show up in distance (the straight-line form of the Pythagorean theorem), in standard deviation (the square root of variance), and in any formula that undoes a square. This page does the root only. It does not solve an equation. For the spread of a list of numbers, use the standard deviation calculator, which takes the root of the variance for you.