# Exponents

Section P.2

**Exponential Notation
for Positive Integers **

Let b be a real number or a variable

representing a real number. Let n be a

positive integer.

where there are n b’s

Example:

Example:

**Product Rule for Exponents **

What happens to the exponents when

we multiply

Solution :

**Exponential Rules**

We want the Product Rule for

Exponents to hold for
powers that are

not positive integers as well. Determine

the definition of each
of the following

exponential rules by ensuring that the

Product Rule for
Exponents holds.

For any real number a ,
what does

Solution

We want to define a zero power so that

What must a^{0}equal for this to happen?

For any real number a,
what does

Solution

We want to define a negative power so that

What must equal for this to happen?

so a negative power means

reciprocal

**Division Rule for Exponents**

For any non-zero real number a, what

does

Solution

Using the rule for negative exponents

So when we divide expressions with the

same base, we subtract powers.

**Other Exponential Rules**

Determine the rules for the following by

expanding the
expression .

**Skills**

Use the Exponential Rules to simplify

the following
expressions to a common

form having

Only positive exponents

All like terms combined

Constant portion reduced to lowest terms

**Skills Practice**

How do we extend the notion of exponents to

the rational numbers? What is

Solution:

Using the Power to a power rule, examine

So the power 1/n undoes

the power n. What operation undoes
taking

an nth power?

Example: a^{2} can be undone by taking
that

is . ,So we define

**Radical Rules**

Simplifying expressions that have

radicals can be done
by converting the

radical to a rational power and then

applying the exponential
rules. Try one

of these three examples.

Common simplified form for radical

expressions is

All factors removed from radical

Index of radical reduced to lowest terms

Rationalize the denominator (no radical left

in the denominator)

**Skills Practice**

Simplify the following radical expression

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