indicate raising to a power by:

Superscripting in typographic formulas, such as

Using ** in plain text formulas, such as

Using ^ in plain text formulas, such as

All the preceding expressions are read as “five to the third power,” “five to the power of three,” or

“five cubed.” It says to multiply three fives together: 5 × 5 × 5, which gives you 125.

Here are some other features of power:

A power doesn’t have to be a whole number. You can raise a number to a fractional power (such

as 3.8). You can’t visualize this in terms of repeated multiplications, but your scientific calculator

can show you that

is equal to approximately 37.748.

A power can be negative. A negative power indicates the reciprocal of the quantity, which is

when you divide the quantity by 1 (meaning

). So

means 1 divided by x, and in general,

is the same as

(such as 2^–3 = ½).

Remember the constant e (2.718…)? Almost every time you see e used in a formula, it’s being raised

to some power. This means you almost always see e with an exponent after it. Raising e to a power is

called exponentiating, and another way of representing

in plain text is exp(x). Remember, x doesn’t

have to be a whole number. By typing =exp(1.6) in the formula bar in Microsoft Excel (or doing the

equation on a scientific calculator), you see that exp(1.6) equals approximately 4.953. We talk more

about exponentiating in other book sections, especially Chapters 18 and 24.

Taking a root

Taking a root involves asking the power question backwards. In other words, we ask: “What base

number, when raised to a certain power, equals a certain number?” For example, “What number, when

raised to the power of 2 (which is squared), equals 100?” Well,

(also expressed

) equals

100, so the square root of 100 is 10. Similarly, the cube root of 1,000,000 is 100, because

(also expressed

) equals a million.

Root-taking is indicated by a radical sign (√) in a typeset formula, where the term from which we

intend to take the root is located “under the roof” of the radical sign, as 25 is shown here:

. If no

numbers appear in the notch of the radical sign, it is assumed we are taking a square root. Other roots

are indicated by putting a number in the notch of the radical sign. Because

is 256, we say 2 is the

eighth root of 256, and we put 8 in the notch of the radical sign covering 256, like this:

. You also

can indicate root-taking by expressing it different ways used in algebra:

is equal to

and can be

expressed as

in plain text.

Looking at logarithms

In addition to root-taking, another way of asking the power question backwards is by saying, “What

exponent (or power) must I raise a particular base number to in order for it to equal a certain

number?” For root-taking, in terms of using a formula, we specify the power and request the base. With

logarithms, we specify the base and request the power (or exponent).

For example, you may ask, “What power must I raise 10 to in order to get 1,000?” The answer is 3,