Squares, Square Roots, Cubes, and Cube Roots

A square root undoes squaring: √x asks for the nonnegative number that, multiplied by itself, equals x. Because any real number squared is zero or positive, there's no real number you can square to get a negative result — so sqrt of a negative doesn't exist in the real numbers. A cube root undoes cubing, and cubing keeps the sign of its input: a negative number cubed stays negative, a positive stays positive. That means every real number, positive or negative, has a real cube root. This asymmetry between even and odd roots is the whole story behind why one radical refuses negative inputs and the other doesn't.

Simplifying a square root means hunting for a perfect-square factor hiding inside. √200 becomes √(100 times 2), and since √100 = 10, that pulls out to 10 times √2. The same move works with variables: √(9x²) splits into √9 times √(x²), giving 3x when x is positive. This connects directly to the rest of the algebra on the exam — quadratic formula answers, distance-formula results, and rational expressions all tend to leave you with a radical that needs this same cleanup before you can match it to an answer choice.

Squares and square roots also hide a subtle trap once variables enter the picture, which is why √(x²) is not automatically x — it's the absolute value of x, since the root always returns a nonnegative number regardless of what x was.