Simplifying Exponential
Expressions
Exponential Notation
Exponent
Base
Base raised to an exponent
Example: What is the base and exponent of the
following expression?
2 is the
7 is the
base
7
2
exponent
Goal
To write simplified statements that contain
distinct bases, one whole number in the
numerator and one in the denominator,
and no negative exponents.
Ex:
9 a b
1

4 3 2
 6a b

2 1 2 2
c
8 4
9b c

12
4a
Multiplying Terms
When we are multiplying terms, it is easiest
to break the problem down into steps.
First multiply the number parts of all the
terms together. Then multiply the variable
parts together.
Examples:
Only the z is
squared
2
.
.
a. ( 4x )( -5x ) = ( 4 -5 )( x x ) = -20x
b. (5z2)(3z)(4y) = (5.2.3.4)(y.z.z) = 120yz2
Exploration
Evaluate the following without a calculator:
34 = 81
33 = 27
32 =
1
÷3
÷3
9
÷3
3 = 3
Describe a pattern and find the answer for:
÷3
0
3 = 1
Zero Power
0
a =1
Anything to
the zero
power is one
Can “a” equal zero?
No.
You can’t divide by 0.
Exploration
Simplify:
Use the
definition of
exponents to
expand
x x
3
4
There are 7
“x” variables
xxx xxxx
3 4
x
7
x
Notice (from
the initial
expression)
3+4 is 7!
Product of a Power
If you multiply
powers having the
same base, add the
exponents.
a
mn
Example
Simplify:
Add the
exponents
since the bases
are the same
x  3 x  y
2
x
9
29
3 1
11
3x
0
Anything
raised to the
0 power is 1
Practice
Simplify the following expressions:
1) x  x  x
5
6
2) 2  x  z  3  x  6x
0
3
3)  9 x y
3
5
2
 4 x y 
2
5
4
 36x y
5
9
Exploration
x 
3
3
3
3
3
x x x x x
35
x
15
x
Simplify:
Adding 3 five times is
equivalent to multiplying
3 by 5. The same
exponents from the initial
expression!
5
3
The Product
of a Power
Rule says to
add all the 3s
Use the
definition of
exponents to
expand
Power of a Power
a
mn
To find a power of a power,
multiply the exponents.
Example
Simplify:
Multiply the
powers of a
exponent
raised to
another power
Any base
without a
power, is
assumed to
have an
exponent of 1
2s  s
 t 
2 6
3 3
4t
2
2s s t 4t
2
26
33
2 s  s  t  4t
1
12
9
112 9  2
2  4s
t
13 11
8s t
2
Multiply
numbers
without
exponents and
add the
exponents
when the bases
are the same
Practice
Simplify the following expressions:
4
2
1)  y
y
2)  a   a   a
3) x  y  x  x  y
17
8
2
4
2
5
5
3
4
23
6
2
x y
11
Exploration
 z x
2
Simplify:
The Product of
a Power Rule
says to add the
exponents with
the same bases
5
z xz xz xz xz x
2
Adding 2 five times
is equivalent to
multiplying 2 by 5
Notice: Both the z2
and x were raised
to the 5th power!
2
2
25 5
z x
5 10
x z
2
2
Use the
definition of
exponents to
expand
Power of a Product
a b
m
If a base has a
product, raise each
factor to the power
m
Example
Simplify:
Everything
inside the
parentheses is
raised to the
exponent
outside the
parentheses
 3x 
2
 2 xy 
4 5
 2  x y
2
5 20
9 x  32 x y
2  5 20
9  32 x y
 3 
2
x
5
2
288x y
7
20
5
Multiply the
powers of a
exponent raised to
another power
45
Multiply
numbers
without
exponents and
add the
exponents
when the bases
are the same
Practice
Simplify the following expressions:
1)   pqr    p q r
5
2)  2ab
3)
5 5 5
  2a 
 512a b
- 2 x  3x yz 
 54x y z
4
2
5
3
19 8
2
3
4
7
3 12
First Four
1. 125x3
2. 64d6
3. a7b7c
4. 64m6n6
5. 100x2y2
6. -r5s5t5
7. 27b4
8. -4x7
9. -15a5b5
10. r8s12
11. 36z11
12. 18x5
13. 4x9
14. a4b4c6
15. 125y12
16. 64x11
17. 256x12
18. 9a8
19. 729z10
20. 321
21. 108a11
22. -81x17
Exploration
55
Complete
the tables
(with
fractions)
by finding
the pattern.
3125
54
625
53
125
52
25
51
5
50
1
5-1
1/5
5-2
5-3
5-4
1/25
1/125
1/625
÷5
÷5
÷5
÷5
÷5
÷5
÷5
÷5
÷5
1
25
1
24
1
23
1
22
1
21
1
20
1
2 1
1
2 2
1
2 3
1
2 4
1/32
1/16
1/8
¼
½
1
2
4
8
16
x2
x2
x2
x2
x2
x2
x2
x2
x2
Negative Powers
1
m
a
A simplified
expression has
no negative
exponents.
Negative
Exponents “flip”
and become
positive
1
m

a
m
a
Example
Simplify:
All of the old
rules still apply
for negative
exponents
10 3
4a b 5a
4  5a
4
10  4 3
b
6 3
Flip ONLY the
thing with the
negative exponent
to the bottom and
the exponent
becomes positive
20a b
20b
6
a
3
This is not
simplified since
there is a
negative
exponent
Example
Simplify:
Everything with
a positive
exponent stays
where it is.
2
12 x y
3
8x
1
2 3
12 x x
1
8y
423
12 x
8y
4
3 x5
2y
Everything with a
negative exponent
is flipped and
exponent becomes
positive.
Since all of the
negative
exponents are
gone, apply all
of the old rules
to simplify.
Practice
Simplify the following expressions:
1) 8
3

1
512
2
6x
2)
5 3
4x y
3
 x7
6
a
b  2
4b
8
3) 3 x y x
4)  2a
3

3 x7
3
2y
8
3y
4
2
Exploration
10
Simplify:
Use the
definition of
exponents to
expand
x
6
x
x x x x  x  x  x  x  x  x
x x x x  x  x
The 6 “x”s in the
denominator cancel 6
out of the 10 “x”s in the
numerator. This is the
same as subtracting
the exponents from the
initial expression!
10  6
x
x
4
Since
everything is
multiplied,
you can
cancel
common
factors
Only 4 “x”s
remain in the
numerator
Quotient of a Power
a
mn
To find a quotient of a power,
subtract the denominator’s
exponent from the
numerator’s exponent if the
bases are the same.
a0
Example
6
2x y
2 3
6x y
Simplify:
Divide the
base
numbers first
Not simplified
since there is a
negative
exponents
1
2
6
x
1
3
6 2
4
y
x y
x4
3 y2
13
Subtract the
exponents of
the similar
bases since
there is division
2
Flip any
negative
exponents
Practice
Simplify the following expressions:
6 0
3
ab
a
1)
3  5
5a
6
5
12 x
3x
2)
12
12 
y
4 xy
9
14 x y
3)
3
4x y
3

6 2
7x y
2
Exploration
6
Simplify:
Use the
definition of
exponents to
expand
a
b
a
 
b
a a
b b
    
Use the definition of
exponents to rewrite.
Notice: Both the
numerator and
denominator were
raised to the 6th power!
a
b
a
b
a a a a a a
bbbbbb
6
a
6
b
a
b
Multiply the
fractions
Power of a Quotient
m
a
m
b
To find a power of a quotient,
raise the denominator and
numerator to the same power.
Example
2
Simplify:
Everything in
the fraction is
raised to the
power out side
the parentheses.
 3   2x y
    5
 y  x
2

3
2
y
y2
2
3
Subtract the
exponents
when there is
division, and
add when there
is multiplication
2

7
23
3



3
2 x y
53
x
73
8 x6 y 21
Multiply the
fractions
15
x
2
6 21
y 8x y
15
9x
221
8y
156
9x

8y
23
9 x9
Practice
Simplify the following expressions:
 a 
1)  0 
 bc 
3
8
 2 x 
2)  2 
 y 

4
a
b8
8
 16 x4
s f5
3) 

4
 zr 


2
24
y
7
35
 r 28s14 z7
f