Algebra2go®
Objective 1
Exponents
Understand the Relationship between
Multiplication and Exponents
Recall that 3 ⋅ 4represents an addition problem.
3⋅ 4 = 3 +3 +3 +3.
But what about the expression
3 ⋅3 ⋅3 ⋅3?
This is where the exponent is used!
Exponents are used to represent repetitive
multiplication of a quantity.
Using an exponent of 4, we write 3 ⋅ 3 ⋅ 3 ⋅ 3 as 34
where 3 is called the base and 4 is the exponent.
Base
34
Suppose we are given
which is equal to 8.
If we are given
23 . This represents 2 ⋅ 2 ⋅ 2
x3 , this represents
So what do you think
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Exponent
x ⋅ x ⋅ x.
x3 ⋅ x4 is equal to?
Algebra2go®
Well since x3 = x ⋅ x ⋅ x and x4 = x ⋅ x ⋅ x ⋅ x
we get the following result.
x3 ⋅ x4 = x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x = x 7
Notice how we can just add the exponents.
x3 ⋅ x4 = x3+4= x 7
When multiplying two quantities that have the
same base, we can add the exponents.
xa ⋅ xb = xa
+b
Suppose we are given x + x + x + x . Notice that
we have four x’s being added together.
Recall that multiplication is used to represent
repetitive addition of the same quantity.
Using the Commutative Property for
multiplication we can now show that
4 ⋅ x = x ⋅ 4 = x + x + x + x = 4x
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Notice that
and
4x = x + x + x + x
x4 = x ⋅ x ⋅ x ⋅ x .
Algebra2go®
Note:
x3 is said “x raised to the third power” or
“x cubed”.
x2 is said “x raised to the second power”
or “x squared”.
Answer the following homework questions.
In Exercises 1 – 9, write each quantity in expanded form.
Recall: 4x=x+x+x+x and x4 = x ⋅ x ⋅ x ⋅ x .
1) 5 2
4) a4
7) ab2
2) y2
5) 3a
8) x 2y2
3) 4y
6) 2x
9) p3q 4
In Exercises 10 – 18, add or multiply as indicated, if possible.
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10) 3x+4x
13) a4 ⋅ a3 ⋅ a2
16) 2h + 2b
11) y3 ⋅ y3
14) w ⋅ w 2
17) h2 + b2
12) 3y+3y
15) w + 2w
18) 2c ⋅ c 2
Algebra2go®
Objective 2
Understand the Zero Exponent
What is the value of 2 ?
We can arrive at a general conclusion by
noticing the following pattern.
4
2 = 2 ⋅ 2 ⋅ 2 ⋅ 2 = 16 Here the numbers
Notice that the
23 = 2 ⋅ 2 ⋅ 2 = 8 are being divided
2
exponents are
by the base 2 as
⋅
2
2
4
=
=
2
decreasing by 1!
you move
1
downward!
2
=2
2=
1
=1
20 =
0
Let’s now try this with a base of 3.
4
3 = 3 ⋅ 3 ⋅ 3 ⋅ 3= 81
Notice that the
33 = 3 ⋅ 3 ⋅ 3 = 27
2
exponents are
3 ⋅3 = 9
decreasing by 1! 3 =
3
=3
31 =
1
=1
30 =
Here the numbers
are being divided
by the base 3 as
you move
downward!
Regardless of what base you use (except for 0),
you will always get the zero power to equal 1!
A base of 0 does not work because we can never
0
divide by zero! 0 is undefined!
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Algebra2go®
Answer the following homework questions.
In Exercises 19 – 27, find the value of each expression!
Note: In every problem you must first evaluate the quantity with
the exponent before performing the arithmetic operation!
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19) 2 3 + 32
22) 34 ⋅ 1 12
25) 2 3 ⋅ 32
20) 32 − 12 0
23) 0 10 ÷ 7 2
26) 11 2 − 8 2
21) 43 ÷ 2 3
24) 102 ÷ 03
27) 5 0 + 40