Transcript Chapter 2

2.1 Simplifying Expressions
• Term: product or quotient of numbers, variables,
and variables raised to powers
5
2
3
3z
x, 15 y ,  2 , xz
x y
2
• Coefficient: number before the variables
If none is present, the coefficient is 1
• Factors vs. terms:
In “5x +y”, 5x is a term.
In “5xy”, 5x is a factor.
2.1 Simplifying Expressions
• When you read a sentence, it split up into words.
There is a space between each word.
• Likewise, an is split up into terms by the +/-/=
sign: 2
2
1
1
3
x 
2
x

6
y  3
• The only trick is that if the +/-/= sign is in
parenthesis, it doesn’t count:
2
3
x  3

1
2
x 
1
6
x  3
2.1 Simplifying Expressions
• Like Terms: terms with exactly the same variables
that have the same exponents
• Examples of like terms:
5 x and  12 x
3 x 2 and 5x 2
• Examples of unlike terms
2 y and  7 y
x and 2 y
2
2.1 Simplifying Expressions
• Combining Like Terms: the distributive
property allows you to combine like terms
• Examples of combining like terms:
5 x  (  12 x)  (5  12) x  7 x
3x  5 x  (3  5) x  8 x
2
2
2
2
2.2 The Product Rule and Power
Rules for Exponents
• Review: PEMDAS (order of operations) –
note that exponentiation is number 2.
• Product rule for exponents:
a a
n
• Example:
m
5 5  5
2
3
a
23
nm
5
5
2.2 The Product Rule and Power
Rules for Exponents
• Power Rule (a) for exponents:
a 
m n
 a nm
• Power Rule (b) for exponents:
ab 
m
a b
m
• Power Rule (c) for exponents:
a
 
b
m
m
a
 m
b
m
2.2 The Product Rule and Power
Rules for Exponents
• A few tricky ones:
 2   2  2  2  8
3
3
 2  2   2  2  2  8
4
 2   2  2  2  2  16
4
4
 2  2   2  2  2  2  16
3
2.2 The Product Rule and Power
Rules for Exponents
• Examples (true or false):
t t  t
4
3
12
( t 4 ) 3  t 12
s  t 
2
s  t 
3
 s t
3
3
 s2  t 2
2.2 The Product Rule and Power
Rules for Exponents
• Formulas and non-formulas:
 a  b n  a  n  b  n (distributive property)

a  b n  a n  b n
 a  b n  a n  b n ,
 a  b 2  a 2  b 2 ,

a 2  b2  a  b
( power rule b)
a  b n  a n  b n
a  b 2  a 2  b 2
2.3 Integer Exponents
• Definition of a zero exponent:
a 0  1 (no matter wha t a is)
• Definition of a negative exponent:
a
n
1
1
 n  
a
a
n
2.3 Integer Exponents
• Changing from negative to positive exponents:
a m
bn
 m
n
b
a
• This formula is not specifically in the book but is
used often:
p
p
m
n
a

 bn





b

 am





2.4 The Quotient Rule
• Quotient rule for exponents:
m
a
mn
a
n
a
2.4 The Quotient Rule
• Examples (true or false):
0
  10   1
1  2
0
2
3
2
x
x
 2
2
y
y
23
5

2
22
2.4 The Quotient Rule
• Putting it all together (example):
3xy 2 x y 
2
3
2
 3xy2 23 x 6 y 3
1 6
 3(8) x
 24 x y
7
5
y
23
2.4 The Quotient Rule
• Another example:
3
 2x y 
 3 xy 

   2 1 
2 
 3 xy 
 2x y 
3 3 6
3 x y
27 3 6 6  3
 3 6 3  8 x y
2 x y
2
1
9
27
y
27 3 9
8 x y 
3
8x
2
3
2.5 Scientific Notation
•
•
A number is in scientific notation if :
1. It is the product of a number and a 10 raised to a
power.
2. The absolute value of the first number is between 1
and 10
Which of the following are in scientific notation?
– 2.45 x 102
– 12,345 x 10-5
– 0.8 x 10-12
– -5.2 x 1012
2.5 Scientific Notation
•
Writing a number in scientific notation:
1. Move the decimal point to the right of the first nonzero digit.
2. Count the places you moved the decimal point.
3. The number of places that you counted in step 2 is the
exponent (without the sign)
4. If your original number (without the sign) was
smaller than 1, the exponent is negative. If it was
bigger than 1, the exponent is positive
2.5 Scientific Notation
• Converting to scientific notation (examples):
6200000  6.2 10?
.00012  1.2 10?
• Converting back – just undo the process:
6.203 1023  620,300,000,000,000,000,000,000
1.86 105  186,000
2.5 Scientific Notation
• Multiplication with scientific notation (answers
given without exponents):
4 10  5 10   4  5 10
5
8
5
10 8

 20 10 3  .02
• Division with scientific notation:
4 10   4  10
5 10  5 10
12
12
4
4
 80,000,000
 .8 1012 4  .8 108