Monomials are algebraic expressions consisting of a number and variables raised to powers. There are three key rules for manipulating monomials: 1) When multiplying monomials, add the exponents of like variables. 2) When raising a monomial to a power, multiply the coefficient by itself and multiply each variable exponent by the outer power. 3) These rules allow complex monomial expressions to be simplified through exponent operations.

1. Monomials
Multiplying Monomials and Raising
Monomials to Powers

2. Vocabulary
• Monomials - a number, a variable, or a product of a
number and one or more variables
• 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials.
• Constant – a monomial that is a number without a
variable.
• Base – In an expression of the form xn, the base is x.
• Exponent – In an expression of the form xn, the
exponent is n.

3. Writing - Using Exponents
Rewrite the following expressions using exponents:
x x x x y y
The variables, x and y, represent the bases. The
number of times each base is multiplied by itself will
be the value of the exponent.
4 2
x x x x y y x y

4. Writing Expressions without
Exponents
Write out each expression without exponents (as
multiplication):
3 2
8a b 8 a a a b b
4
xy xy xy xy xy
or
x x x x y y y y

5. Product of Powers
Simplify the following expression: (5a2)(a5)
There are two monomials. Underline
them.
What operation is between the two
monomials?
Multiplication
!
Step 1: Write out the expressions in expanded form.
2 5
5a a 5 a a a a a a a
Step 2: Rewrite using exponents.
2 5 7 7
5a a 5 a 5a

6. Product of Powers Rule
For any number a, and all integers m and n,
am • an = am+n.
9 4 13
1) a a a
2 10 12
2) w w w
5 6
3) r r r
5 3 8
4) k k k
2 2 2 2
5) x y x y

7. Multiplying Monomials
If the monomials have coefficients, multiply
those, but still add the powers.
9 4 13
1) 4a 2a 8a
2 10 12
2) 7w 10w 70w
5 6
3) 2r 3r 6r
5 3 8
4) 3k 7k 21k
2 2 2 2
5) 12 x 2y 24x y

8. Multiplying Monomials
These monomials have a mixture of
different variables. Only add powers of like
variables.
9 3 4 13 4
1) 4a b 2a b8a b
2 5 10 2 12 7
2) 7w y 10 w y 70w y
3 5 6 3
3) 2rt 3r 6r t
5 4 3 3 3 8 4 7
4) 3k mn 7k m n 21k m n
2 3 2 3 5
5) 12 x y 2xy 24x y

9. Power of Powers
Simplify the following: ( x3 ) 4
The monomial is the term inside the
parentheses.
Step 1: Write out the expression in expanded form.
3 4 3 3 3 3
x x x x x
x x x x x x x x x x x x
Step 2: Simplify, writing as a power.
3 4 12
x x
Note: 3 x 4 = 12.

10. Power of Powers Rule
n
m mn
For any number, a, and all integers m and n, a a .
9 10 90
1) b b
3 3 9
2) c c
12 2 24
3) w w

11. Monomials to Powers
If the monomial inside the parentheses has a
coefficient, raise the coefficient to the power, but
still multiply the variable powers.
9 3 27
1) 2b 8b
3 3 9
2) 5c 125c
12 2 24
3) 7w 49w

12. Monomials to Powers
(Power of a Product)
If the monomial inside the parentheses has more
than one variable, raise each variable to the outside
power using the power of a power rule.
(ab)m = am•bm
3 2 4 4 3 4 4 2 4
5w xy 5 w x y
34 4 24
625 w x y
12 4 8
625w x y

13. Monomials to Powers
(Power of a Product)
Simplify each expression:
9 4 3 27 12
1) 2b c 8b c
5 3 3 15 9
2) 5a c 125a c
12 4 2 24 8 2
3) 7w y z 49w y z