Exponents are those funny little numbers (or sometimes letters; see Algebra page) floating above larger numbers or letters, e.g.:

2^{3}

Don’t worry. Exponents are very powerful (pun fully intended) tools, and they are very straightforward to understand.

The larger number is called the “base.” The exponent simply means that you multiply the base by itself the indicated number of times, e.g.:

2^{3} = 2 x 2 x 2 = 8

Exponents = shorthand for multiple multiplications = all back to basic arithmetic, Q.E.D. (sensing a pattern yet?)

The way an exponential expression is said aloud is the base to the power of the exponent, e.g. “2 to the 3^{rd} power.” However, we apply special terminology when the exponent is 2 or 3, as discussed below.

When the exponent is 2, we generally say the base number “squared” instead of “to the 2^{nd} power.” This is illustrated in the figure below. The area of a rectangle is given by the length times the width. A square is a special kind of rectangle for which the length and the width are equal, and they are given by “a” in the figure. The area of a square is:

a x a = a^{2} (“a squared”)

When you raise a number to the 2^{nd} power, “a” in this example, it is equivalent to calculating the area of a square with sides of length a. Thus raising a number to the 2^{nd} power is called “squaring” the number. In this case, we would say “a squared.”

When the exponent is 3, we generally say the base number “cubed” instead of “to the 3^{rd} power.” This is illustrated in the figure below. The volume of a cube is given by the length times the width times the height, all of which are given by “a” in this example. Thus the volume is:

a x a x a = a^{3} (“a cubed”)

When you raise a number to the 3^{rd} power, “a” in this example, it is equivalent to calculating the volume of a cube with sides of length a. Thus raising a number to the 3^{rd} power is called “cubing” the number. In this case, we would say “a cubed.”

Taking the “square root” of a number is simply the reverse of squaring a number. It is finding the number whose square is the number in question. For example, what is the square root of 4?

Well…

2 x 2 = 2^{2} = 4

Thus 2 squared = 4, which means that the square root of 4 is 2.

Similarly, the square root of 9 is 3, because 3^{2} = 9.

The square root of 16 is 4, because 4^{2} = 16.

And so on.

Similarly, taking the “cube root” of a number is finding the number whose cube is the number in question.

The cube root of 8 is 2, because 2^{3} = 8.

The cube root of 27 is 3, because 3^{3} = 27.

Etc., etc., etc.

This it is, and nothing more.