# A Math Primer

## Exponents and Roots

A. Exponents

**Definition
**In x

^{n}, x is the base, and n is the exponent (or power)

We defined positive integer powers by

x

^{n}= x · x · x · . . . · x (n factors of x)

**Properties**

The above definition can be extended by requiring

other powers (i.e. other than positive integers) to

behave like the positive integer powers. For example,

we know that

for positive integer powers, because we can write out

the multiplication.

Example:

x

^{2}x

^{5}= (x · x)(x · x · x · x · x)

=

x · x · x · x · x · x · x = x

^{7}

We now require that this rule hold even if n and m

are not positive integers, although this means that we

can no longer write out the multiplication (How do

you multiply something by itself a negative number

of times? Or a fractional number of times?).

We can find several new properties of exponents by

similarly considering the rule for dividing powers :

This rule is quite reasonable when m and n are

positive integers and m > n. For example:

where indeed 5 – 2 = 3.

However, in other cases it leads to situations where

we have to define new properties for exponents.

First, suppose that m < n. We can simplify it by

canceling like factors as before :

But following our rule would give

In order for these two results to be consistent, it must

be true that

or, in general,

Notice that a minus sign in the exponent does not

make the result negative—instead, it makes it the

reciprocal of the result with the positive

exponent.

Now suppose that n = m. The fraction becomes

which is obviously equal to 1. But our rule gives

Again, in order to remain consistent we have to say

that these two results are equal, and so we define

x^{0} = 1

for all values of x (except x = 0, because 0^{0} is

undefined)

** Summary of Exponent Rules**

The following properties hold for all real numbers x,

y, n, and m, with these exceptions:

1. 0^{0} is undefined

2. Dividing by zero is undefined

3. Raising negative numbers to fractional powers

can be undefined

**B. Roots
Definition**

Roots are the inverse of exponents. An nth root

“undoes” raising a number to the nth power, and

vice-versa. (The correct terminology for these types

of relationships is inverse functions, but powers and

roots can only be strictly classified as inverse

functions if we take care of some ambiguities

associated with plus or minus signs, so we will not

worry about this yet). The common example is the

square root , which “undoes” the act of squaring. For

example, take 3 and square it to get 9. Now take the

square root of 9 and get 3 again. It is also possible to

have roots related to powers other than the square.

The cube root , for example, is the inverse of raising

to the power of 3. The cube root of 8 is 2 because 2

^{3}

= 8. In general, the nth root of a number is written:

if and only if

because 4^{3} = 64

We leave the index off the square root symbol only

because it is the most common one. It is understood

that if no index is shown, then the index is 2.

if and only if

because 4^{2} = 16

**Square Roots
**The square root is the inverse function of squaring

(strictly speaking only for positive numbers, because

sign information can be lost)

**Principal Root**

Every positive number has two square roots, one

positive and one negative

**Example:** 2 is a square root of 4 because 2 x 2 = 4,

but –2 is also a square root of 4 because (–2) x

(–2) = 4

To avoid confusion between the two we **define** the

symbol (this symbol is called a radical ) to
mean

the **principal ** or** positive ** square root.

The convention is: For any positive number x,

is the positive root, and

is the negative root.

If you mean the negative root, use a minus sign in

front of the radical.

**Example:**

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