The document reviews the basic rules for exponents that were covered in Algebra 1, including: - When multiplying like variables, you add their exponents - When dividing like variables, you subtract their exponents - When an exponent is raised to another exponent, you multiply the exponents - When a variable has a negative exponent, you change its position in a fraction and the exponent becomes positive The document then provides examples of applying multiple exponent rules to evaluate multi-step algebraic expressions involving variables with exponents.

1. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n

2. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
- When you multiply like variables you ADD
their exponents

3. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
- When you multiply like variables you ADD
their exponents
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
- When you divide like variables you
SUBTRACT their exponents

4. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
- When you multiply like variables you ADD
their exponents
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
- When you divide like variables you
SUBTRACT their exponents
- When an exponent is raised to another
exponent, you MULTIPLY exponents

5. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
- When you multiply like variables you ADD
their exponents
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
- When you divide like variables you
SUBTRACT their exponents
- When an exponent is raised to another
exponent, you MULTIPLY exponents
- When a variable has a negative exponent,
you change its position in a fraction and the
exponent becomes positive.

6. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
x 2 ⋅ x 5 = x 2+5 = x 7
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
- When you divide like variables you
SUBTRACT their exponents
- When an exponent is raised to another
exponent, you MULTIPLY exponents
- When a variable has a negative exponent,
you change its position in a fraction and the
exponent becomes positive.

7. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
x 2 ⋅ x 5 = x 2+5 = x 7
x8
= x 8−5 = x 3
x5
- When an exponent is raised to another
exponent, you MULTIPLY exponents
- When a variable has a negative exponent,
you change its position in a fraction and the
exponent becomes positive.

8. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
x 2 ⋅ x 5 = x 2+5 = x 7
x8
= x 8−5 = x 3
x5
(x )
3 4
= x 3⋅4 = x12
- When a variable has a negative exponent,
you change its position in a fraction and the
exponent becomes positive.

9. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
x 2 ⋅ x 5 = x 2+5 = x 7
x8
= x 8−5 = x 3
x5
(x )
3 4
x
−3
= x 3⋅4 = x12
1
= 3
x

10. Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
a ⋅a = a
m
n
m+n
m
a
= a m−n
n
a
(a )
m n
a
−n
=a
1
= n
a
m⋅ n
x 2 ⋅ x 5 = x 2+5 = x 7
x8
= x 8−5 = x 3
x5
(x )
3 4
x
−3
= x 3⋅4 = x12
1
= 3
x
What we are going to do in Pre – Calc is take these rules and combine them into
multi – step problems that might have all four rules utilized.

11. Algebraic Expressions – Rules for Exponents
Example # 1 :
( 2a ) ( − 4a 3b 2 )(b 2c )

12. Algebraic Expressions – Rules for Exponents
Example # 1 :
( 2a ) ( − 4a 3b 2 )(b 2c )
= ( 2 ⋅ ( − 4) ) ( a ⋅ a 3 )( b 2 ⋅ b 2 )( c )
= −8 ⋅ a1+3 ⋅ b 2+ 2 ⋅ c
Since everything is multiplied, we can multiply all
integers, and all like variables.

13. Algebraic Expressions – Rules for Exponents
Example # 1 :
( 2a ) ( − 4a 3b 2 )(b 2c )
= ( 2 ⋅ ( − 4 ) ) ( a ⋅ a 3 )( b 2 ⋅ b 2 )( c )
= −8 ⋅ a1+3 ⋅ b 2+ 2 ⋅ c
= −8a 4b 4 c

14. Algebraic Expressions – Rules for Exponents
Example # 1 :
( 2a ) ( − 4a 3b 2 )(b 2c )
= ( 2 ⋅ ( − 4 ) ) ( a ⋅ a 3 )( b 2 ⋅ b 2 )( c )
= −8 ⋅ a1+3 ⋅ b 2+ 2 ⋅ c
= −8a 4b 4 c
Example # 2 :
m x + 6 ⋅ m −3 x ⋅ n 2 x −5 ⋅ n −3 x +9

15. Algebraic Expressions – Rules for Exponents
Example # 1 :
( 2a ) ( − 4a 3b 2 )(b 2c )
= ( 2 ⋅ ( − 4 ) ) ( a ⋅ a 3 )( b 2 ⋅ b 2 )( c )
= −8 ⋅ a1+3 ⋅ b 2+ 2 ⋅ c
= −8a 4b 4 c
Example # 2 :
m x + 6 ⋅ m −3 x ⋅ n 2 x −5 ⋅ n −3 x +9
Don’t PANIC !!! The rule for multiplying like variables still
applies. Exponents can be algebraic, integers, or fractions

16. Algebraic Expressions – Rules for Exponents
Example # 1 :
( 2a ) ( − 4a 3b 2 )(b 2c )
= ( 2 ⋅ ( − 4 ) ) ( a ⋅ a 3 )( b 2 ⋅ b 2 )( c )
= −8 ⋅ a1+3 ⋅ b 2+ 2 ⋅ c
= −8a 4b 4 c
Example # 2 :
m x + 6 ⋅ m −3 x ⋅ n 2 x −5 ⋅ n −3 x +9
=m
( x + 6 ) + ( −3 x )
⋅n
( 2 x −5 ) + ( −3 x + 9 )
We will still ADD
exponents
Don’t PANIC !!! The rule for multiplying like variables still
applies. Exponents can be algebraic, integers, or fractions

17. Algebraic Expressions – Rules for Exponents
Example # 1 :
( 2a ) ( − 4a 3b 2 )(b 2c )
= ( 2 ⋅ ( − 4 ) ) ( a ⋅ a 3 )( b 2 ⋅ b 2 )( c )
= −8 ⋅ a1+3 ⋅ b 2+ 2 ⋅ c
= −8a 4b 4 c
Example # 2 :
m x + 6 ⋅ m −3 x ⋅ n 2 x −5 ⋅ n −3 x +9
= m ( x + 6 ) + ( −3 x ) ⋅ n ( 2 x −5 ) + ( −3 x +9 )
=m
−2 x+6
n
We will still ADD
exponents
− x+4
Now just treat the exponent like an algebraic expression
where we combine like terms…

18. Algebraic Expressions – Rules for Exponents
Example # 3 :
[ 4( − 2 x y ) ]
2
3
3 2
When you have imbedded parentheses and
exponents outside, start with the innermost set and
work your way “out”.

19. Algebraic Expressions – Rules for Exponents
Example # 3 :
[ 4( − 2 x y ) ]
3
3 2
2
[(
= 4 ( − 2) ⋅ x
2
2⋅2
⋅y
3⋅2
)]
3
Evaluate this first, applying the exponent outside to
ALL of the terms inside.

20. Algebraic Expressions – Rules for Exponents
Example # 3 :
[ 4( − 2 x y ) ]
2
[(
3
3 2
= 4 ( − 2) ⋅ x
[(
2
= 4 4x y
4
6
)]
2⋅2
3
⋅y
3⋅2
)]
3

21. Algebraic Expressions – Rules for Exponents
Example # 3 :
[ 4( − 2 x y ) ]
= [ 4( ( − 2 ) ⋅ x
2
3
3 2
2
[(
)]
= [16 x y ]
= 4 4x y
4
4
6
2⋅2
⋅y
3⋅2
)]
3
3
6 3
I like to multiply what is inside the brackets before I
apply the outside exponent.

22. Algebraic Expressions – Rules for Exponents
Example # 3 :
[ 4( − 2 x y ) ]
= [ 4( ( − 2 ) ⋅ x
2
3
3 2
2
[(
)]
= [16 x y ]
= 4 4x y
4
4
6
2⋅2
⋅y
3⋅2
)]
3
6 3
= 163 ⋅ x 4⋅3 y 6⋅3
Apply the outside exponent.
3

23. Algebraic Expressions – Rules for Exponents
Example # 3 :
[ 4( − 2 x y ) ]
2
3
3 2
[(
= 4 ( − 2) ⋅ x
2
[(
)]
= [16 x y ]
= 4 4x y
4
4
6
⋅y
3⋅2
)]
3
3
6 3
4⋅3
= 16 ⋅ x y
3
2⋅2
= 163 x12 y18
6⋅3
or 4096 x12 y18

24. Algebraic Expressions – Rules for Exponents
Example # 4 :
3a 3b −2
− 6ab −5
There are two ways to attack this problem.
1. Apply the division rule using negative
exponents.
2. Use the negative exponent rule first, then
apply the division rule.

25. Algebraic Expressions – Rules for Exponents
Example # 4 :
3a 3b −2
− 6ab −5
There are two ways to attack this problem.
1. Apply the division rule using negative
exponents.
2. Use the negative exponent rule first, then
apply the division rule.
3 3−1 − 2−( −5 )
=
⋅ a ⋅b
−6
1 2 3
=− a b
2

26. Algebraic Expressions – Rules for Exponents
Example # 4 :
3a 3b −2
− 6ab −5
There are two ways to attack this problem.
1. Apply the division rule using negative
exponents.
2. Use the negative exponent rule first, then
apply the division rule.
3a 3b 5
=
− 6ab 2
I applied the negative exponent rule and moved
any variable with a negative exponent to the
other part of the fraction and made the exponent
positive…

27. Algebraic Expressions – Rules for Exponents
Example # 4 :
3a 3b −2
− 6ab −5
There are two ways to attack this problem.
1. Apply the division rule using negative
exponents.
2. Use the negative exponent rule first, then
apply the division rule.
3a 3b 5
=
− 6ab 2
3 3−1 5− 2
=
⋅ a ⋅b
−6
1 2 3
=− a b
2
Apply the normal division of like variables rule…

28. Algebraic Expressions – Rules for Exponents
Example # 5 :
(36 x y )
1
2
2
(x y )
3
4
1
3
1
2
This problem has a few rules to apply,
apply the exponent to each
parentheses first…

29. Algebraic Expressions – Rules for Exponents
Example # 5 :
(36 x y )
1
2
2
(x y )
3
=
4
1
2
This problem has a few rules to apply,
apply the exponent to each
parentheses first…
1
3
1
2
36 ⋅ x
3⋅ 1
3
2⋅ 1
2
x ⋅y
⋅y
4⋅ 1
3
1⋅1
2 2
=
6 xy
xy
4
3
1
4
When you have an integer
to the ½ power, it is the
square root of the integer.

30. Algebraic Expressions – Rules for Exponents
Example # 5 :
(36 x y )
1
2
2
(x y )
3
=
4
1
2
This problem has a few rules to apply,
apply the exponent to each
parentheses first…
1
3
1
2
36 ⋅ x
2⋅ 1
2
⋅y
3⋅ 1
3
4⋅ 1
3
1−1
1−4
4 3
x ⋅y
= 6⋅ x
⋅y
1⋅1
2 2
=
6 xy
xy
4
3
1
4
Now you divide your
variables…

31. Algebraic Expressions – Rules for Exponents
Example # 5 :
(36 x y )
1
2
2
(x y )
3
=
4
1
2
This problem has a few rules to apply,
apply the exponent to each
parentheses first…
1
3
1
2
36 ⋅ x
2⋅ 1
2
⋅y
3⋅ 1
3
4⋅ 1
3
1−1
1−4
4 3
x ⋅y
= 6⋅ x
= 6y
13
− 12
⋅y
=
6
y
13
12
1⋅1
2 2
=
6 xy
xy
1
4
4
3
Anything to the zero
power = 1, and it is not
necessary to change
improper fractions to
mixed numbers…

32. Algebraic Expressions – Rules for Exponents
Example # 6 :
 − 12 x y z

 4 x 3 y −1 z − 2

−2
3 −5




−1

33. Algebraic Expressions – Rules for Exponents
Example # 6 :
 − 12 x y z

 4 x 3 y −1 z − 2

−2
3 −5




−1
On this one I would reduce any integer
fraction first…

34. Algebraic Expressions – Rules for Exponents
Example # 6 :
 − 12 x y z

 4 x 3 y −1 z − 2

−2
3 −5
 − 3x y z
=  3 −1 − 2
 x y z

−2




3 −5
−1




On this one I would reduce any integer
fraction first…
−1

35. Algebraic Expressions – Rules for Exponents
Example # 6 :
 − 12 x y z

 4 x 3 y −1 z − 2

−2
3 −5




−1
−1
 − 3x y z 
=  3 −1 − 2 
 x y z



 ( − 3) −1 x ( − 2 )( −1) y ( 3 )( −1) z ( −5 )( −1)
=

x ( 3 )( −1) y ( −1)( −1) z ( − 2 )( −1)

 ( − 3) −1 x 2 y −3 z 5 

=
−3 1 2


x yz


−2
3 −5




Now apply the negative
exponent outside…

36. Algebraic Expressions – Rules for Exponents
Example # 6 :
 − 12 x y z

 4 x 3 y −1 z − 2

−2
3 −5




−1
−1
 − 3x y z 
=  3 −1 − 2 
 x y z



 ( − 3) −1 x ( − 2 )( −1) y ( 3 )( −1) z ( −5 )( −1) 

=
( 3 )( −1) ( −1)( −1) ( − 2 )( −1)


x
y
z


 ( − 3) −1 x 2 y −3 z 5   x 2 x 3 z 5 
=
=
−3 1 2
 − 3 y1 y 3 z 2 



x yz


 
−2
3 −5
Move your negative
exponent variables…

37. Algebraic Expressions – Rules for Exponents
Example # 6 :
 − 12 x y z

 4 x 3 y −1 z − 2

−2
3 −5




−1
−1
 − 3x y z 
=  3 −1 − 2 
 x y z



 ( − 3) −1 x ( − 2 )( −1) y ( 3 )( −1) z ( −5 )( −1) 

=
( 3 )( −1) ( −1)( −1) ( − 2 )( −1)


x
y
z


 ( − 3) −1 x 2 y −3 z 5   x 2 x 3 z 5 
=
=
−3 1 2
 − 3 y1 y 3 z 2 



x yz


 
−2
3 −5
x 2 + 3 z 5− 2
x5 z 3
=
=
1+ 3
− 3y
− 3y4
Apply multiplication and
division of variables
rules…