Exponent Study Guide
Laws of Exponents and Exponent Facts
The Basics
Exponential Notation – Repeated MULTIPLICATION
Example: 3 × 3 × 3 × 3 × 3 = 35
An Exponent is a number that shows you how
many times the number is to be used in
multiplication.
It is written as a small number to the right
and above the base number.
When the base is either a FRACTION or a
NEGATIVE, you must use PARENTHESES.
Multiplication of Exponents – Product Law
**The bases must be the SAME**
LAW
xm ∙ xn = x m + n
When you Multiply numbers with exponents, you
keep the common base and ADD the EXPONENTS
Example: 35 × 37
= (3 × 3 × 3 × 3 × 3) × (3 × 3 × 3 × 3 × 3 × 3 × 3)
=3×3×3 ×3×3×3×3×3 ×3×3×3×3
= 3 5 + 7 = 312
Power Raised to a Power
LAW
(xm)n
=
xm ∙ n
When you have a POWER RAISED TO A POWER,
you MULTIPLY the EXPONENTS
Example: (53)4 = (53)(53 )(53)(53)
= (5 × 5 × 5) × (5 × 5 × 5) × (5 × 5 × 5) × (5 × 5 × 5)
=5×5×5 ×5×5×5×5×5 ×5×5×5×5
= 512 = 53 × 4
Negative Exponents
LAW
x-n
=
1
𝑥𝑛
When you have a number with a NEGATIVE
EXPONENT, you DROP THE BASE AND THE
EXPONET DOWN TO THE DENOMINATOR AND
MAKE THE EXPONENT POSITIVE
Expanded Form of the Decimal in Exponential
Notation
629.125
600 + 20 + 9 + .1 + .02 + .005
6 (100) + 2 (10) + 9 (1) + 1 (.1) + 2 (.01) + 5 (.001)
(6 × 102) + (2 × 101) + (9 × 100) + ( 1 × 10 -1) + (2 ×
10-2) + (5 × 10-3)
COUNT YOUR ZEROS – That will give you your
exponent
FIND THE ONES PLACE– That will always be 100
because anything raised to the zero power is 1
Product of Numbers with Exponents
Positive Base – The Product will ALWAYS be POSITIVE
Negative Base – ONLY if the base is NEGATIVE
Use the Even/Odd Exponent Rule
Negative Base/Even Exponent – Positive Product
Negative Base/Odd Exponent – Negative Product
Example: Does (–3)7 have a positive or negative
product?
Is the Base Positive or Negative? – Negative
Even or Odd? – Odd
NEGATIVE
Division of Exponents – Quotient Law
LAW
**The bases must be the SAME**
𝑥𝑚
𝑥𝑛
= x m–n
When you divide numbers with exponents, you keep
the common base and SUBTRACT the EXPONENTS
𝟒𝟗
Example: 𝟓
𝟒
= 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 45
4×4×4×4 ×4
Power Raised to a Power (FRACTION)
(xy)n = xnyn
𝑥 𝑛
(𝑦) =
LAW
𝑥𝑛
𝑦𝑛
When you have a QUANTITY RAISED TO A POWER,
you must RAISE EACH TERM IN THAT QUANTITY TO
THAT POWER
(xm ∙ yp)n = (xm)n ∙ (yp)n
𝑥𝑚
𝑛
( 𝑦𝑝 ) =
(𝑥𝑚 )𝑛
(𝑦𝑝 )𝑛
Facts
ANYTHING raised to the zero power is equal to 1
74850 = 1
(a3b 5c15)0 = 1
ANYTHING raised to the 1 power is equal to itself
74851 = 7485
(a 3b5c15)1 = a3b 5c15
Multiplication is COMMUTATIVE – This means you
can CHANGE THE ORDER – Group your numbers
together and your variables together when
multiplying
When simplifying numbers in exponential notation,
all exponents should be positive and in one fraction
(if applicable)
Exponent Study Guide
APPLICATION OF THE EXPONENT LAWS
3
7
Example 1) (2x ) (17x )
The operation between each term is multiplication
2 · x3 · 17 · x7
Multiplication is commutative, therefore, you can
change the order
2 · 17 · x3 · x7
Use product law to simplify
34 · x3 + 7 = 34x10
Example 3)
35
27
Must have the same base in order to use the
division law. It is easier to make the bigger number
smaller rather than the smaller number bigger.
27 must have a base of 3
3
1
3 =3=3
32 = 3 × 3 = 9
33 = 3 × 3 × 3 = 𝟐𝟕
Replace 27 with the equivalent 33
35
33
Use quotient law
35−3 = 32
Example 5)
(𝑥 4 𝑦𝑧 5 )2
This really means
(𝑥 4 𝑦1 𝑧 5 )(𝑥 4 𝑦1 𝑧 5 )
Since multiplication is commutative, we can
change the order
𝑥 4 𝑥 4𝑦 1𝑦 1𝑧 5 𝑧 5
We can then write this as
(𝑥 4 )2 (𝑦 1)2 (𝑧 5 )2
Use power to a power law
𝑥 4 × 2𝑦 1 × 2𝑧 5 × 2 = 𝐱 𝟖 𝐲𝟐 𝐳𝟏𝟎
Example 7)
𝑥 3 𝑦 −8
Since the two terms do not have the same base,
you CANNOT use the product law. However, you
can use the negative exponent law and make it in
to one fraction.
𝑥3
1
𝐱𝟑
× 8= 𝟖
1
𝑦
𝐲
Example 2)
(11𝑥 16 )
6
𝑥4
Put (11𝑥 16 ) over 1 in order to multiply two fractions
together
(11𝑥16 )
1
6
· 4
𝑥
Multiply the numerators together, multiply the
denominators together
6 (11𝑥 16 )
6 ∙ 11 ∙ 𝑥 16
66𝑥 16
=
=
𝑥4
𝑥4
𝑥4
Use quotient law
66 ∙ 𝑥 16−4 = 𝟔𝟔𝒙𝟏𝟐
Example 4)
162 ∙ 26
Must have the same base in order to use the
multiplication law. It is easier to make the bigger
number smaller rather than the smaller number bigger.
162 must have a base of 2
(16 )2
21 = 2 = 2
2
2 =2 ×2=4
23 = 2 × 2 × 2 = 8
24 = 2 × 2 × 2 × 2 = 𝟏𝟔
Replace 16 with the equivalent 2 4
(24)2 ∙ 26
Use power to a power law
2 4 × 2 ∙ 26 = 28 ∙ 26
Use product law
2 8 + 6 = 2 14
Example 6) 𝑥 5 𝑥 −6
Use the product law
𝑥 5+(−6)
Integer Rules – Addition – Same Signs Add, Same Signs
Keep the Signs; Different Signs Subtract, Different Signs
Keep the Sign of the Larger Absolute Value –
Subtraction – Keep Change Change then Use Addition
Rules
𝑥 −1
Use negative exponent law – drop it to the
denominator and make the exponent positive
1
𝟏
=
1
𝑥
𝐱
𝑥3 𝑦7
Example 8)
𝑥12
Since the operation between x and y is multiplication,
you can separate the fractions
𝑥3
𝑦7
×
𝑥 12
1
Use the quotient law (and Integer Rules!)
𝑥 3−12 𝑦7 = 𝑥 −9𝑦 7
Positive Exponents ONLY (Negative Exponent Law)
1
𝑦7
𝐲𝟕
×
=
𝑥9
1
𝐱𝟗