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
