Root

The nth root of a number is the value that, when raised to the power of n, gives back that original number; in other words, taking a root reverses raising to a power. The small number n, called the index, tells you how many times the result must be multiplied by itself: the second root is the familiar square root, and the third root is the cube root. Roots of positive numbers are always positive, while even roots of negative numbers have no real value because no real number multiplied by itself an even number of times can be negative. Odd roots, by contrast, work for negative numbers too, since the sign is preserved. Most roots are irrational, meaning their decimal expansion never ends or repeats, so they are usually left in symbolic form or approximated. Roots appear throughout mathematics and science, from computing distances and areas with the Pythagorean theorem to standard deviation in statistics, growth rates in finance, and countless formulas in physics and engineering. Understanding them is essential because so many real-world relationships involve squares, cubes, and their inverses.