Download Using the Quotient Rule and Defining the Zero

Math 64
Appendix C "Quotient Rule and Negative Exponents"
Objectives:
* Use the quotient rule for exponents, and de…ne a number raised to the 0 power.
* Simplify expressions containing negative exponents.
* Use the rules and de…nitions for exponents to simplify exponential expressions.
Using the Quotient Rule and De…ning the Zero Exponent
Let’s study pattern for simplifying exponential expressions that involves quotients.
Quotient Rule for Exponents:
If m and n are positive integers and a is a real number, then
Example 1: (Using the quotient rule)
Simplify each quotient:
y 10
a)
y6
c)
12a4 b11
ab
b)
511
58
d)
7x2 y 6
14x2 y 3
Zero Exponent:
a0 = 1; as long as a is not 0.
Example 2: (Using the zero exponent)
Simplify each expression.
a) 60
b) ( 8)
0
c)
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80
d) 7y 0
Notes by Bibiana Lopez
Prealgebra by Elayn Martin-Gay
Appendix C
Simplifying Expressions Containing Negative Exponents
Our work with exponential expressions so far has been limited to exponents that are positive integers or 0. Here we will
also give meaning to an expression like x
5
9
or y
x
5
:
Negative Exponents:
If a is a real number other than 0 and n is an integer, then
and
Example 3: (Simplifying expressions containing negative exponents)
Simplify by writing each expression with positive exponents only.
a) 5
d)
2
6
7
b) 5x
2
e)
2
c) 4
x
x 4
f)
1
+3
1
y 4
y6
Simplifying Exponential Expressions
All the previously stated rules for exponents apply for negative exponents also. Here is a summary of the rules and
de…nitions for exponents.
Summary of Exponents Rules:
If m and n are integers and a; b; and c are real numbers, then
1: Product Rule
5: Quotient Rule
2:Power Rule
6: Zero Exponent:
3:Power of a Product
7: Negative Exponent:
4:Power of a Quotient
Page: 2
Notes by Bibiana Lopez
Prealgebra by Elayn Martin-Gay
Appendix C
Example 4: (Simplifying exponential expressions)
Simplify each expression. Write each result using positive exponents only.
a) y
6
y3 y
c)
3y 9 z 10
e)
8m5 n
4
2y 3 z
1
b)
12
7m2 n
a6 b
4
d) x8 x
4
f)
Page: 3
2x10 y
3 8
a
11
3
b
x
2
9x4 y
7
Notes by Bibiana Lopez