exponents. what we want is to see the child in pursuit of knowledge, and not knowledge in pursuit of...
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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.
34
Base
Exponent
Definition of Exponent
An exponent tells how many times a number is multiplied by itself.
34 Exponent
What an Exponent Represents
An exponent tells how many times a number is multiplied by itself.
34= 3 x 3 x 3 x 3
times
4
How to Read an Exponent
This exponent is read:
three to the fourth power
34
Base
Exponent
Common Exponents
This exponent is read:
three to the second power
orthree squared
32 Exponent
Common Exponents
This exponent is read:
three to the third power
orthree cubed
33 Exponent
What is the Exponent?
2 x 2 x 2 = 23
What is the Base and the Exponent?
8 x 8 x 8 x 8 = 8 4
How to Multiply Out an Exponent (Standard Form)
= 3 x 3 x 3 x 33
927
81
4
Write in Standard Form
4 2= 16
Rules of Exponents
Exponents come with their own set of rules
Rules follow a natural emerging pattern
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
Multiplication Rule
7 times
Words Numbers Algebra
To multiply powers with the same base, keep the base and add the exponents.
bm × bn
= bm + n
35 × 38 = 35 + 8 = 313
MULTIPLYING POWERS WITH THE SAME BASE
Multiplication Rule
ExamplesMultiply and write the product as one power:
66 × 63
69
66 + 3
45 × 47
41245 + 7
Add exponents.
Add exponents.
Multiplication Rule
244 × 244
25 × 2
2 6
25 + 1
248
24 4 + 4
Think: 2 = 2 1
Add exponents.
Add exponents.
Multiplication RuleAdditional ExamplesMultiply and write the product as one power:
Notice what occurs when you divide powers with the same base.
55
53=
5 × 5 × 5
5 × 5 × 5 × 5 × 5
= 5 × 5
= 52
= 5 × 5 × 5
5 × 5 × 5 × 5 × 5
Division Rule
DIVIDING POWERS WITH THE SAME BASE
Words Numbers Algebra
To divide powers with the same base, keep the base and subtract the exponents.
6569 – 469
64
= = bm – nbm
bn
=
Division Rule
Subtract exponents.
72
75 – 3
75
73
Divide and w Write the product as one power
210
29
Subtract exponents.210 – 9
2 Think: 2 = 21
Division Rule
When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0 exponent.
This result can be confirmed by writing out the factors.
1 = 42
4242 – 2 = 40 = 1
=
=
(4 × 4)(4 × 4)
= 11
1 =42
2= (4 × 4)
4 (4 × 4)
Zero Rule
THE ZERO POWER
Words Numbers Algebra
The zero power of any number except 0 equals 1.
1000 = 1
(–7)0 = 1a0 = 1, if a 0
Zero Rule
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 + 4 x 2
Solution: Student 1 3 + 4 x 2
= 7 x 2 = 14
Student 2 3 + 4 x 2 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 Evaluation Operation
6 + 7 x 8 = 6 + 7 x 8 Multiplication
= 6 + 56 Addition
= 62
16 ÷ 8 - 2 = 16 ÷ 8 - 2 Division
= 2 - 2 Subtraction
= 0
(25 - 11) x 3 = (25 - 11) x 3 Parentheses
= 14 x 3 Multiplication
= 42
Time to do some computing!
Evaluate using the order of operations.
3 + 6 x (5 + 4) ÷ 3 - 7
Solution:
Step 1: = 3 + 6 x 9 ÷ 3 - 7 Parentheses
Step 2: = 3 + 54 ÷ 3 - 7 Multiplication
Step 3: = 3 + 18 - 7 Division
Step 4: = 21 - 7 Addition
Step 5: = 14 Subtraction
1) 5 + (12 – 3) 5 + 9 14
2) 8 – 3 • 2 + 7 8 - 6 + 7 2 + 7 9
3) 39 ÷ (9 + 4) 39 ÷ 13 3
Examples
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
Thus
Evaluating this expression, we get:
Fractions
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.
Write an arithmetic expression
Add Parentheses to Obtain Result
7 − 1 + 6 = 07 − (1 + 6) = 0
1 + 2 × 5 + 6 = 21(1 + 2) × 5 + 6 = 21
2 + 2 × 5 ÷ 3 − 1 = 10(2 + 2) × 5 ÷ (3 − 1) = 10
3 + 8 ÷ 2 = 7
3 + (8 ÷ 2) = 7
8 + 3 − 7 = 48 + 3 − 7 = 4
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.
Summary