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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 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 Prev Next