Transcript Exponents - Dalton State

Exponents
What we want is to see the child in
pursuit of knowledge, and not
knowledge in pursuit of the child.
George Bernard Shaw
EXPONENTS
Exponents represents a mathematical
shorthand that tells how many times a
number is multiplied by itself.
Exponents

Understanding exponents is important
because this shorthand is used throughout
subsequent mathematics courses. It
appears often in formulas used in science,
business, statistics, and geometry.
Location of Exponent

An exponent is a little number high and to
the right of a regular or base number.
Base
3
4
Exponent
Definition of Exponent

An exponent tells how many times a
number is multiplied by itself.
3
4
Exponent
What an Exponent Represents

An exponent tells how many times a number
is multiplied by itself.
4
3 =3x3x3x3
times
How to Read an Exponent

This exponent is read:
three to the fourth power
Base
3
4
Exponent
Common Exponents

This exponent is read:
three to the second power
or
three squared
3
2
Exponent
Common Exponents

This exponent is read:
three to the third power
or
three cubed
3
3
Exponent
What is the Exponent?
2x2x2=2
3
What is the Base and the
Exponent?
8x8x8x8=8
4
How to Multiply Out an Exponent
(Standard Form)
4
3 =3x3x3x3
9
27
81
Write in Standard Form
4
2
= 16
Rules of Exponents

Exponents come with their own set of rules

Rules follow a natural emerging pattern
Multiplication Rule
The factors of a power, such as 74, can be grouped
in different ways. Notice the relationship of the
exponents in each product.
7 × 7 × 7 × 7 = 74
(7 × 7 × 7) × 7 = 73 × 71 = 74
(7 × 7) × (7 × 7) = 72 × 72 = 74
Multiplication Rule
Multiply 34 × 33
34 × 33
= (3 × 3 × 3 × 3) × (3 × 3 × 3)
= (3 × 3 × 3 × 3 × 3 × 3 × 3)
= 37
7 times
Multiplication Rule
MULTIPLYING POWERS WITH THE SAME BASE
Words
To multiply
powers with the
same base, keep
the base and
add the
exponents.
Numbers
35 × 38
= 35 + 8
= 313
Algebra
bm × bn
= bm + n
Multiplication Rule
Examples
Multiply and write the product as one power:
66 × 63
6+3
6
9
6
45 × 47
5+7
4
12
4
Add exponents.
Add exponents.
Multiplication Rule
Additional Examples
Multiply and write the product as one power:
25 × 2
25 + 1
26
Think: 2 = 2 1
Add exponents.
244 × 244
24 4 + 4
24 8
Add exponents.
Division Rule
Notice what occurs when you divide powers
with the same base.
5× 5× 5 × 5× 5
55
=
3
5
5×5×5
5× 5× 5 × 5× 5
= 5×5×5
=5×5
= 52
Division Rule
DIVIDING POWERS WITH THE SAME BASE
Words
To divide powers
with the same
base, keep the
base and subtract
the exponents.
Numbers
69
64
=
69 – 4
=
Algebra
6
5
bm
bn
=
bm – n
Division Rule
Divide and w Write the product as one power
75
73
75 – 3
Subtract exponents.
72
210
29
210 – 9
2
Subtract exponents.
Think: 21 = 2
Zero Rule
When the numerator and denominator have the
same base and exponent, subtracting the
exponents results in a 0 exponent.
2
4
1= 2
4
=
42 – 2 = 40 = 1
This result can be confirmed by writing out the factors.
42
42
=
(4
(4
×
×
4)
4)
=
(4
(4
×
×
4)
4)
=
1
1
=
1
Zero Rule
THE ZERO POWER
Words
Numbers
The zero power of
any number
except 0 equals 1.
1000 = 1
(–7)0 = 1
Algebra
a0 = 1, if a  0
ORDER OF OPERATIONS
How to do a math problem
with more than one operation in
the correct order.
Order of Operations

Problem:
Evaluate the following
arithmetic expression:
3+4x2

Solution:
Student 1
3+4x2
=
7x2
=
14
Student 2
3+4x2
3+8
11
Order of Operations

It seems that each student interpreted the
problem differently, resulting in two different
answers.
 Student 1 performed the operation of addition
first, then multiplication
 Student 2 performed multiplication first, then
addition.
Order of Operations

When performing arithmetic operations
there can be only one correct answer. We
need a set of rules in order to avoid this
kind of confusion. Mathematicians have
devised a standard order of operations for
calculations involving more than one
arithmetic operation.
Order of Operations
Rule 1:
First perform any calculations
inside parentheses.
Rule 2:
Next perform all multiplications and
divisions, working from left to right.
Rule 3:
Lastly, perform all additions and
subtractions, working from left to
right.
Example
Expression
6+7x8
=
=
=
Evaluation
6+7x8
6 + 56
62
Operation
Multiplication
Addition
16 ÷ 8 - 2
16 ÷ 8 - 2
2-2
0
Division
Subtraction
(25 - 11) x 3
14 x 3
42
Parentheses
Multiplication
=
=
=
(25 - 11) x 3 =
=
=
Time to do some computing!
Evaluate using the order of operations.
3 + 6 x (5 + 4) ÷ 3 - 7
Solution:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
=
=
=
=
=
3+6x9÷3-7
3 + 54 ÷ 3 - 7
3 + 18 - 7
21 - 7
14
Parentheses
Multiplication
Division
Addition
Subtraction
Examples
2) 8 – 3 • 2 + 7
8 - 6 +7
2 + 7
9
3) 39 ÷ (9 +
4)
39 ÷ 13
3
1) 5 + (12 – 3)
5+ 9
14
Fractions
Evaluate the arithmetic expression below:
This problem includes a fraction bar, which means we
must divide the numerator by the denominator. However,
we must first perform all calculations above and below
the fraction bar BEFORE dividing.
The fraction bar can act as a grouping symbol
Fractions
Thus
Evaluating this expression, we get:
Write an arithmetic expression
Mr. Smith charged Jill $32 for parts and $15 per
hour for labor to repair her bicycle. If he spent 3
hours repairing her bike, how much does Jill owe
him?
Solution:
32 + 3 x 15
32 + 3 x 15
=
= 32 + 45 = 77
Jill owes Mr. Smith $77.
Add Parentheses to Obtain Result
7−1+6=0
7 − (1 + 6) = 0
3+8÷2=7
3 + (8 ÷ 2) = 7
1 + 2 × 5 + 6 = 21
(1 + 2) × 5 + 6 = 21
8+3−7=4
8+3−7=4
2 + 2 × 5 ÷ 3 − 1 = 10
(2 + 2) × 5 ÷ (3 − 1) =
10
Summary
When evaluating arithmetic expressions, the order
of operations is:
•Simplify all operations inside parentheses.
•Perform all multiplications and divisions,
working from left to right.
•Perform all additions and subtractions, working
from left to right.
If a problem includes a fraction bar, perform all calculations above and
below the fraction bar before dividing the numerator by the
denominator.