Name _____________________________________________________________________
Chapter 8 Study Guide and Notes
base
37
power
8-1 – Zero and Negative Exponents Pages 430-435
Any base (except 0) taken to the zero power is one.
70 = 1
x0 = 1
(-12)0 = 1
(½)0 = 1
Any base (except 0) can be taken to any negative integer power. The answer will be
the reciprocal of the number taken to the positive of that power.
(3)-2
=
1
32
=
1
9
2 −3
5 3
5
2
( ) =( ) =
125
1
8
7−2
= 72 = 49
8-2 – Scientific Notation Pages 436 -440
Scientific notation takes number-- especially large and small numbers --and puts them
into a format that is easier to use. The front number must be between 1 and 10 [only
one digit (not zero) is before the decimal point.] The second number is ten taken to
an integer power.
2,540,000,000,000 = 2.54 x 1012
12 decimal places to the left
8.32 x 107 = 83,200,000
0.000456 = 4.56 x 10-4
4 decimal places to the right
6.79 x 10-5 = 0.0000679
8-3 Multiplication Properties of Exponents Pages 441-446
To multiply the same bases taken to powers: Add the exponents.
32 ● 35 = 3(2 + 5) = 37
x5 ● x-3 = x(5 +(-3)) = x2
To multiply using scientific notation: multiply the first number of each expression and
add the exponents from the powers of ten.
(3.4 x 109) ● (2.3 x 105) = (3.4 ● 2.3) x (109 ●105) = 7.72 x 1014
In some cases, you may need to adjust, so that the number is still in scientific notation.
(8.55 x 108) ● (9.23 x 1011) = (8.55 ● 9.23) x (108 ●1011) = 78.91.65 x 1019 = 7.891 x 1020
8-4 More Multiplication Properties of Exponents Pages 447 -452
To find any base taken to the power of a power: Multiply the exponents.
(32)3 = 3(2●3) = 36
(-2x3y7z-4)2 = (-2)2x(3●2)y(7●2)z(-4●2) =4 x6y14z-8
Remember to multiply the constants.
(-2x4)3(4y3)-2(3 x-1y0)2 = (-8x12) ●(
1
16𝑦
9
−72𝑥 12
𝑥
16𝑥 2 𝑦6
6 ) ● ( 2) =
=
−9𝑥 12
2𝑥 2 𝑦 6
8-5
Division Properties of Exponents
Pages 453-459
To divide the same bases taken to powers: Subtract the exponents.
58
56
−6𝑥 6 𝑦 −3
= 5(8-6) = 52 = 25
3𝑥 3 𝑦 −2
= -2x(6-3)y(-3-(-2)) = -2x3y-1 =
−2𝑥 3
𝑦
When taking a fraction or an expression that includes a division bar to a power
remember to raise both the numerator and denominator to the power.
(
3𝑥 3 𝑦 5
4𝑥 5 𝑧
2
10 ) =
9𝑥 (3●2) 𝑦 (5●2)
16𝑥 (5●2) 𝑧 (10●2)
=
9𝑥 6 𝑦 10
16𝑥 10 𝑧 20
=
9𝑦 10
16𝑥 4 𝑧 20
8-6 Geometric Sequences Pages 460 - 465
Geometric Sequences are formed by repeated multiplication by a common ratio.
Term Number
1
12
2
36
3
108
4
324
5
972
6
2916
3
Common ratio
x3
x3
x3
x3
x3
x3
This geometric sequence: -2 ● (2) n looks like:
36
12
=
108
36
324
972
=108 = 324 = 3
The formula for this
sequence would be:
4●3n
9
-3, - 2 , -
27
, 4
81
8
, ...
8-7 Exponential Functions Pages 468-473
To evaluate an exponential expression use the numbers given for each variable.
Evaluate f(x) = 7●4x for x = 1, 2, and 3.
f (1) =7 (41) = 7 ● 4 = 28
f (2) =7 (42) = 7 ● 16 = 112
f (3) =7 (43) = 7 ● 64 = 448
To graph an exponential expression, set up a table and evaluate the expression at
several points. Graph those points, and then graph your curve.
3
Graph: y = (− 2 ) ● 2𝑥
x
3
(x,y)
(− ) ● 2𝑥
2
3
3 (-2,- 3)
-2
(− ) ● 2−2 = −
8
2
8
3
3 (-1,- 3)
-1
(− ) ● 2−1 = −
4
2
4
3
3 (0,- 3)
0
(− ) ● 20 = −
2
2
2
1 (− 3 ) ● 21 = − 3 (1,- 3)
2
2
3
(− 2 ) ● 22 = − 6
(2,- 6)
8-8 Exponential Growth and Decay Pages 475-482
Exponential Growth means b is greater than 1. b is the growth factor.
y = a ● bx
a is the value of y when x = 0. a > 0
Exponential Decay means b is less than 1. b is the decay factor.
Compound Interest is an exponential function. You receive interest on your interest.
y = a ● bx With compund interest problems a in your initial investment or principal, b
is your rate of growth ( 100% + interest rate), and x is the time the principal is invested.
If you invest $1000 on the day you turn 18, and withdraw it on your 65 th birthday
assuming a constant interest rate of 4% compounded annually (once a year). You can find
how much you can withdraw by solving:
y = 1000 ● (1.04)47