Chapter 1 Section 8
Exponential Notation
and
Order of Operations
Definition
An exponent is used to indicate repeated
multiplication. The exponent tells how
many times the base is used as a factor.
Example : 23 = 2 ⋅ 2 ⋅ 2
2 is the base, 3 is the exponent
Exponential Notation
For any natural number n,
n factors
n
b = b ⋅ b ⋅ b ⋅ b ⋅ ... ⋅ b
1
Rewrite the following without exponents.
Simplify where appropriate
34
x2
( −2 )
4
− x2
2
(−x )
−24
Evaluate the Following
2
( )
For x = 9: (a) −x ; (b) − x 2
(
For x = 9 and y = 3: x + y
2
)
Bob said Paul is tall.
Bob, said Paul, is tall
2
2 + 3⋅5
Order of Operations Agreement
PEMDAS
•
•
•
•
•
•
•
Parentheses or absolute values first
Exponents
Multiplication or
Division
Addition or
Subtraction
Similar operations are done from left to right.
Order of Operations Agreement
• Remove parentheses or absolute values,
work from the innermost symbols first.
• Exponents
• Multiplication or Division
• Addition or Subtraction
• Similar operations are done from left to
right.
3
When solving problems with a
fraction bar, simplify the numerator
and denominator first, then do the
division last. The fraction bar acts as a
set of parentheses For example:
3(14) − 4
23 − 6
Equivalent Notation
x −1
= (x − 1) / 2
2
x2 − 4
= x2 − 4 / x − 2
x−2
(
)(
)
Your Turn
10 + 5
6 −1
52 + 17
2 ⋅ 5 − 23
(5 − 3)2 + 2
42 − (8 + 2)
4
Definition
Two numbers that are the same distance
away from zero on a number line, but on
opposite sides of zero, are opposites or
negatives.
Definition
If a is any number, the opposite of a is
denoted by
-a
(read the opposite of a)
Example
Find the opposite of each number
•
•
•
•
•
•
-4
3
9
-5
a
x
5
The Double Negative Rule
If a is any number,
-(-a) = a
Example
Simplify each of the following.
•
•
•
•
•
•
-(-8) =
-(-4) =
-(-7) =
-(-s) =
-(-u) =
-(-(-3)) =
Definition
The absolute value of any number is the
distance between the number and zero on
a number line.
|a|
(read the absolute value of a)
6
Absolute Value
• The distance a value is from 0 is called it’s
absolute value.
• We write |a|, read the “absolute value of a”
to represent the units a is from 0.
−10 = 10
10 units
–10
10 units
0
10
Use the order of operations to simplify
3
( ) ( )
2 ⋅ 42 − 7
4 −2 − 5 −6
( 3 − 7 )( 4 − 6 )
2 −5 − 4 − 5
52 − 32
2⋅6 − 4
5 8 − 3⋅ 7
( )
Evaluate the following expressions
7 + x3
when x = 2
−x 2 +x 3 when x = −3
a 3 − 4a
when a = −2
a ( a − 3)
7
Using the STO key
x−
3
x
when x = 5
x 2 + 0.0271x −
x 3 − 4x
(
)
x x−3
11
when x = −3
16
when x = −5
The Mulitplication Property of − 1
− 1⋅ a = −a
Negative one times a is the opposite of a
The opposite of a sum
-1( a + b ) = −a + ( −b )
Distributing the negative over a sum.
− ( a + b ) = −1( a + b )
= −1( a ) + ( −1)( b )
= −a + ( −b )
= −a − b
8
Rewrite the following
without parentheses
− ( 3x + 5)
3a + 2a − ( 4a − 7 )
− ( 6a − 4b )
4x2 + 8 − ( 2x2 − 9)
7 y − ( 2 y + 4)
4 ( x − 3) − 5 ( 2 x − 4 )
9