for Positive Integers
Let b be a real number or a variable
representing a real number. Let n be a
where there are n b’s
Product Rule for Exponents
What happens to the exponents when
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 ,
We want to define a zero power so that
What must a0equal for this to happen?
For any real number a,
We want to define a negative power so that
What must equal for this to happen?
so a negative power means
Division Rule for Exponents
For any non-zero real number a, what
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 .
Use the Exponential Rules to simplify
the following expressions to a common
Only positive exponents
All like terms combined
Constant portion reduced to lowest terms
How do we extend the notion of exponents to
the rational numbers? What is
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
radical to a rational power and then
applying the exponential rules. Try one
of these three examples.
Common simplified form for radical
All factors removed from radical
Index of radical reduced to lowest terms
Rationalize the denominator (no radical left
in the denominator)
Simplify the following radical expression