What a square root is
The square root of a number is the value that, multiplied by itself, gives that number. For example, the square root of 25 is 5, since 5 × 5 = 25. Every positive number has two square roots, one positive and one negative (5 and -5, in the example), but by convention the radical symbol (√) always refers to the positive root, also called the principal root.
Perfect squares
A perfect square is a whole number that results from multiplying another whole number by itself: 1, 4, 9, 16, 25, 36, and so on. The square root of a perfect square is always an exact whole number. Numbers that aren't perfect squares, like 2, 3, 5, or 50, have irrational square roots, with infinite, non-repeating decimals.
How to simplify a radical
Simplifying the square root of a number that isn't a perfect square means rewriting it in its smallest possible form, pulling out any factor that is a perfect square from inside the radical. The process is to find the largest perfect square factor of the number, divide the number by it, and place the root of that factor in front of the remaining radical. For example, 50 = 25 × 2, and since the square root of 25 is 5, the simplified square root of 50 is 5√2.
Worked example
The square root of 72 isn't a whole number, but it can be simplified: 72 = 36 × 2, and the square root of 36 is 6, so the simplified square root of 72 is 6√2, roughly 8.4853 in decimal form.
Square roots of negative numbers
Within the set of real numbers, there is no number that, multiplied by itself, results in a negative value, since a negative number multiplied by another negative always gives a positive result. Because of this, mathematicians created imaginary numbers, based on the imaginary unit i, defined as the square root of -1. The square root of any negative number can then be expressed as a multiple of i: for example, the square root of -16 equals the square root of 16 multiplied by i, or 4i.