Transcript Section 1.8

§ 1.8
Exponents and Order of Operations
Definition of Natural Number Exponent
If b is a real number and n is a natural number,
b  b
b
 b...

b

n
b appears as a
factor n times.
b is the base and n is the exponent.
b n is read “the nth power of b” or “b to the nth power”.
Blitzer, Introductory Algebra, 5e – Slide #2 Section 1.8
Simplifying algebraic expressions that
contain exponents.
Simplify: 3x 2  7 x 2
There are two like terms with same variable factor,
namely x 2 .
3x 2  7 x 2  10 x 2
Simplify: 5 x3  4 x 2 .
This cannot be simplified because they are
not like terms.
Blitzer, Introductory Algebra, 5e – Slide #3 Section 1.8
Simplifying algebraic expressions that
contain exponents.
3x  2 x  5 x  x  (3x  5x )  (2 x  x )  8 x  x
4
2
4
2
4
4
2
2
4
2
In this example, you can see that we combined under addition only
“like terms”. The variable parts of terms must be the same for them
to be “like terms”. Note that we stopped after we had combined
like terms. We can not go farther in simplifying in this example.
Blitzer, Introductory Algebra, 5e – Slide #4 Section 1.8
Order of Operations
1. Perform all operations within grouping symbols.
(Starting with innermost.)
2. Evaluate all exponential expressions.
3. Do all multiplications and divisions in the order
they occur, working left to right.
4. Finally, do all additions and subtractions in the
order in which they occur, working left to right.
Blitzer, Introductory Algebra, 5e – Slide #5 Section 1.8
Examples: Order of Operation
Simplify : 9  2  3
96
Do the multiplica tion first. (2  3  6)
 15
Add.
Simplify : 23  15  5  6
 8  15  5  6 Evaluate the exponentia l expression . (2 3  8)
 836
Perform the division. (15  5  3)
 56
Add and subtract, left to right.
 -1
Blitzer, Introductory Algebra, 5e – Slide #6 Section 1.8
Order of Operations
Example:
Simplify : 5[2(3- 7)  6]
 5[2(-4)  6]
Remove innermost grouping symbol. 3 - 7  - 4
 5[-8  6]
Work inside bracket. Multiply 2  -4  - 8
 5[-2]
Add inside bracket. - 8  6  - 2
 -10
Multiply : 5[-2]  -10
Blitzer, Introductory Algebra, 5e – Slide #7 Section 1.8
Using Order of Operations
Example:
Simplify : 3[5(x - 2)  8]
 3[5x - 10  8]
Remove the innermost grouping symbol
using distributi on. 5(x - 2)  5x - 10.
 3[5x - 2]
Add inside the bracket. - 10  8  - 2
 15x - 6
Use the distributi ve property t o
remove the bracket. 3  5x - 3  2  15x - 6
Blitzer, Introductory Algebra, 5e – Slide #8 Section 1.8
In evaluating expressions, what comes first?
• #1 Start with the
parentheses.
Parentheses say “Me
First!”
• #2 Then evaluate the
exponential
expressions.
• #3 Multiplications and
divisions are equal in
the order of operations
– Perform them next.
• #4 Additions and
subtractions are also
equal to each other in
order – and they come
last.
Remember by “PEMDAS” parentheses, exponents, multiplication, division, addition, subtraction
Blitzer, Introductory Algebra, 5e – Slide #9 Section 1.8
Order of Operations - PEMDAS
Order of Operations
1) First, perform all operations within
grouping symbols
2) Next, Evaluate all exponential
expressions.
3) Next, do all multiplications and
divisions in the order in which they
occur working from left to right.
4) Finally, do all additions and
subtractions in the order in which they
occur, working from left to right.
Blitzer, Introductory Algebra, 5e – Slide #10 Section 1.8
Order of Operations - PEMDAS
EXAMPLE
Evaluate R 3  26  R 4 for R  3 .
SOLUTION
R  26  R 
3
3  26  3
3
3  23
3
4
27  23
4
4
4
Replace R with 3
Evaluate inside parentheses first
3
Evaluate 3 – first exponent
Blitzer, Introductory Algebra, 5e – Slide #11 Section 1.8
Order of Operations - PEMDAS
CONTINUED
4
Evaluate 3 – second exponent
27-2(81)
27-162
Multiply
-135
Subtract
Blitzer, Introductory Algebra, 5e – Slide #12 Section 1.8
Order of Operations
EXAMPLE
Simplify.
4  32
6
2
6
SOLUTION
4  32
6
2
6
6
49
2
6
6
36
2
6
662
2
Evaluating exponent
Multiply
Divide
Subtract
Blitzer, Introductory Algebra, 5e – Slide #13 Section 1.8