# Exponents

I. Counting Factors and Laws of Exponents

II. Negative Exponents

III. Rational Exponents

IV. Evaluating Logs

I. Counting Factors and Laws of Exponents

What is the purpose of an exponent? The expression b n indicates that
b (called the base) is a factor n times.

Illustration:

x4 means x is a factor 4 times or x· x· x· x
(x − 2)10 means (x − 2) is a factor 10 times.

To "simplify", express the total number of factors with a single exponent.

 Example How factor is counted x is a factor (4 + 3) times x is a factor (25 − 10) times x3is a factor 4 times so x is a factor (3 ×4) times xy is a factor 3 times so x AND y are factors 3 times

Based on these examples, we now state the Laws of Exponents.

 Rather than just memorize, think of these laws as a quick way to count factors. Once that happens, the laws become logical and errors are eliminated .

Exercise 1. Express in simplest form:

II. Negative Exponents

If exponents and laws of exponents count factors for positive exponents ,
what does a negative or zero exponent count? This is a hard question to
answer logically so forgive us for just giving definitions.

Def:( )0 = 1 for all ( )

Def:

We are using ( ) instead of a variable .

Illustration:

Keep the following in mind when simplifying expressions with negative ex-
ponents:

1) Exponent laws do not depend on the type of exponent.

just like

2) In a simplified answer all exponents are positive.

Examples: Express in simplest form.

Solutions .

 Applying exponent laws before manipulating to change negative ex- ponents is the simplest approach.

Exercise 2. Express in simplest form.

III. Rational Exponents

Rational ( meaning fractional ) exponents obey the same laws of exponents.

Illustration:

Exercise 3. Simplify.

IV. Evaluating Logs

No study of exponents is complete without some mention of logarithms.
The expression in words is read as "log of b to the base a = x".
What does this mean?

Using blanks, is equivalent to

Illustration:

means Thus   since .
means . Thus since

Exercise 4. Evaluate the following.

Beginning of Topic 108 Skills Assessment

Express in simplest form:

h) Already in simplest form. Sorry for trying to trick you.

Express in simplest form.

Simplify.