Transcript Section 8-1: Zero and Negative Exponents

Section 8-1: Zero and Negative
Exponents
Objectives: (1) To understand that numbers in
the real number system are represented in a
variety of equivalent forms
(2) To add, subtract, multiply and divide
exponential expressions
Properties
• Zero Exponent Property
– For every nonzero number a, a0 = 1
• Examples
– 50 = 1
(-2)0 = 1
(-1.02)0 = 1
(1/2)0 = 1
• Negative Exponent Property
– For every nonzero number a and integer n,
1
-n
a =a
n
• Examples
– 6-4 =
1
64
(-8)-1 =  1
8
Converting Negative Exponents to
Positive Exponents
• Step 1: Make every number with a negative
exponent a fraction (easiest way to convert
any number into a fraction is putting it over 1)
• Step 2: Move the number to the other side of
the fraction bar and change the sign
• Step 3: If there are no numbers left on one
side of the fraction bar, put a 1 to represent
that there was a number there originally.
Examples of Converting Negative
Exponents
• 4-3
• 1_
43
• 1 _____
4•4•4
• 1__
64
This is what we call
simplifying a power
Simplifying an Exponential Expression
Simplify the expression
4yx-3
1
4 y 3 
x 
4y
X3
• Use the definition of
negative exponents
• Simplify
• We’re done!
• Tada!
Simplifying an Exponential Expression
Simplify the expression:
1
w-4
w4
• Move the w to the
other side of the
fraction bar
• You’re done!
• Tada!
Evaluating Exponential Expressions
Evaulate:
3m2t-2 for t = -3, m = 2
3m2
t2
3(2)2
(-3)2
3(4)
9
12
9
4
3
• 1st: Convert to positive
exponents
• 2nd: Plug in values (put them
in parenthesis)
• 3rd : Simplify
• You’re done
Real-World Problem Solving
• Population Growth:
A biologist is studying green peach aphids (a bug).
In the lab, the population doubles every week.
The expression 1000 • 2w models an initial
population of 1000 insects after w weeks of
growth.
a. Evaluate the expression for w = 0. Then describe
what the value of the expression represents in the
situation.
b. Evaluate the expression for w = -3. Then describe
what the value of the expression represents in the
situation.
Part A
1000 • 2w
1000 • 20
1000 • 1
1000
• Plug in 0 for w
• Now evaluate
• What does 1000
represent?
• The initial number of
aphids
Part B
1000 • 2w
1000 • 2-3
1000 • 13
2
1000 •
125
1
8
• Plug in -3 for w
• Now evaluate
• What does 125
represent?
• The number of aphids 3
weeks ago