Download Chapter 2 The Rule of Exponents

Chapter 2 The Rule of Exponents
Objective: Basic rules of exponents, integer exponents,
quotient rule, and scientific notation
1
2.1 Exponential Expressions (Text 2.1A and 2.1B)

An exponential expression of the form an , where a any real number and n is a
positive integer, is equivalent to aaaa (there are n of as). We say a is the
base and n is the exponent. Evaluate: (a) 24 , (b) ( 2)4 , 24

Rules for operations with exponential expression: If m, n and p are positive
integers, then
1.
Product rule am an =a m + n
2.
Basic power rule (am)n = amn
3.
Power rule for a product (ab) n= an bn
4.



;n (ambn ) p=mamp bnpmp
a n a
(
Power rule for a quotient If b 0, then b )  b n
(
a p a
)  np
n
b
b
;
The simplest form of an exponential expression is when it contains only
positive exponents
Make sure follow the order of operations always
2 5
Examples: simplify  4(2a 2b3 )4 ; 3(5k 2 m3 ) 2
; (5t 3) 6
(2 x )


HW2: Pp 84 – 85, #3 – 93 EOO, #95, 97
HW3: Pp 87, #3-43 EOO
2
2.2 Real Numbers/Variables Bases(Text 2.2A and 2.2B)


Zero as an exponent: if a  0, then a0 = 1; Evaluate: 70 , (3)0 , 20 , 70(2)
Negative exponent: if n is any integer and a0 and b0, then a n  ( 1 ) n  ( 1 ) n
a
a
a n
b n ; evaluate : 4 3 ; ( 4 ) 3 ; ( 1 ) 4 ; (7) 3
( ) ( )
b







3
a
3
Zero as an exponent to bases containing variables: Evaluate: t0 , (5k)0 ,
(2s)0 , s0(2)
Reminder: The simplest form of an exponential expression is when it contains
only positive exponents
4 3
Rewrite an exponential expression with positive exponents y2 ; ( ) ;5a 3
n
If m and n are integers, and a and b are nonzero real numbers, then
Rewrite
a 8
b 3
;
3 p 5
q3
;
2 x 5 y 2
3 z 3
; (
a m
bn
 m
b n
a
2 x 2
)
3y2
HW4: P 90, #5 – 39 ODD
HW5: P 93, #1 – 45 ODD
3
am
 a mn
n
a
75
c5
7 3 ; c 3
2.3 Quotient Rule

Examples: simplify

Summary:




33
23
32
; 37 ; 2  7 ; 2  4
;
5s 4t 5
s 3t 2
;
15a 4b 6
5a 3b 2
m n
a a  a
Product rule
Basic power rule (a m ) n  a mn
(ab) n  a nb n
Power rule for a product
a n an
Power rule for a quotient ( )  n
m
n
b
b
m
Quotient rule

Changing from negative to positive exponent
0
Zero as an exponent, if a 0, then a  1
Negative number as an exponent , if a 0 then




a
 a mn
n
a

6 2
3 2
Examples: (3a ) (2a )
HW6: P99 #1 – 61 EOO
15a 4b 6  2
(
)
5a 3b 2
a n 
1
an
6a 4 b 6  2
(
)
3
2
3a b
s 4  s 5
s7
4
2.4-1 Scientific Notation

Scientific notation is a different form to express the decimal number in the
form of a  10n, where n is an integer (positive or negative), and 1  |a| < 10;
it is used to express very large or a very small numbers.

Rule to write scientific notation: if the number is > 10



Move the decimal point to the left of the first digit

The exponent n is positive and equal to the number of decimal places that moved

E.g. 980,000 = 9.8  105 ; 3,500,000; 93,000,000,000
Rule to write scientific notation: if the number is < 1

Move the decimal point to the right of the first nonzero digit

The exponent n is negative and equal to the number of decimal places that moved

E.g. 0.0002= 2  10-4 ; 0.000000016; 0.000000000086
To convert from scientific notation to decimal notation

If the exponent n is positive, move the decimal point to the right n places

If the exponent n is negative, move the decimal point to the left |n| places (same as you add
|n|-1 zeros in front of the decimal point)

E.g. 3.3107; 7.4 108 ; 2.710-5 ; 3.5410-6
5
2.4 – 2Applications

To simplify the big numbers,

Write in scientific notation

Use the rules of exponents to simplify the operation.
2,400,000,000  0.0000063
0.00009  480
(2)
5,600,000  0.000000081
900  0.000000028

Simplify (1)

Example 1: How many miles does light travel in 1 day? The speed of light is
186,000 mi/s. Write the answer in scientific notation.

Example 2: A computer can do an arithmetic operation in 1  10-7 s. How
many arithmetic operations can the computer perform in 1 min? Write the
answer in scientific notation.

HW7: Pp104-106, #1 – 73 EOO

Read p. 107 – 108 summary

Extra credit: Pp109-p110: #5ae, 7ad, 9de, 11eg, 24cd, 15ag, 17ac, 19, 21, 25,
27, 33, 39, 41; p111 chapter 2 test. #1 – 28 all. (bonus 4 points, do all)
6