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# 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.

Zero Exponent

For any real number a , what does

Solution
We want to define a zero power so that

What must a0equal for this to happen?

Negative Exponent

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

Rational Exponents

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: a2 can be undone by taking that
is . ,So we define

Simplifying expressions that have
radicals can be done by converting the
to a rational power and then
applying the exponential rules. Try one
of these three examples.

Rationalizing the Denominator