CHAPTER 2 Overcoming Mathophobia: Reading and Understanding Mathematical Expressions 21

»

» ^{A power can be negative. A negative power indicates the reciprocal of the}

quantity, which is when you divide the quantity by 1 (meaning 1/x). So x ¹

means 1 divided by x, and in general, x

n is the same as 1/x n (such as 2–3 = ½).

Remember the constant e (2.718. . .)? Almost every time you see e used in a for-

mula, 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 e ^x 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 num-

ber?” For example, “What number, when raised to the power of 2 (which is

squared), equals 100?” Well, 10

10 (also expressed 10²) equals 100, so the square

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

100

100

(also expressed 100³) 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: 25 . 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 putt-

ing a number in the notch of the radical sign. Because 2⁸ 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: 256

8

. You also can indicate root-taking by expressing it different ways used

in algebra: x

n

is equal to x

n

1/ and can be expressed as x

n

^ 1/

^{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, because 10

1 000

3

,

. You can say that 3 is the logarithm

of 1,000 (for base 10), or, in mathematical terms: Log10 1 000

3

,

. Simi-

larly, because 2

256

8

, you say that Log2 256

8. And because e^{1 6}

4 953

.

.

,

then Log e 4 953

1 6

.

. .