Monomials are expressions involving numbers and variables with optional exponents. The degree of a monomial is the sum of its variable exponents. There are several laws of exponents for operations involving monomials with exponents, such as multiplying and dividing powers, raising powers to powers, and evaluating powers of products and quotients. Examples demonstrate applying these laws of exponents to simplify expressions involving monomials with exponents.

1. Exponents and
Monomials

2. Monomial is an expression that is a
number, a variable, or a product of a
number and variables.
Constant is a monomial containing no
variables.
Coefficient is a numerical factor of a
monomial.
Degree is the sum of the exponents of a
monomial’s variables.
Power is the expression of the form xn.

3. Laws of Exponents
1
Negative Exponents: a −n
= n
a
1
a
−n =a n
m +n
Multiplying Powers: a g a = a
m
n

4. Laws of Exponents
m
a m −n
Dividing Powers: n =a
a
Power of a Power: (a ) =a
m n mn
( ab )
m
Power of a Product: =a b m m

5. Laws of Exponents
n n
a a
Power of a Quotient:   = n
 b b
−n n
b a
  = bn
a
Power of Zero: a = 1
0

6. Examples using the laws of exponents:
1. (2ab )(−3a b c ) = −6a b c
2 4 2 5 4
2. (6x y2 3
)(−xyz ) = −6x y z
3 4
12
x
3. 4 = x 8
x

7. −10x y 2 4
4. 4 = −2x
5xy
5. (x )= x
2 4
8
6. (−2a b ) = 16a
3 2 4
12 8
b

8. 2
 3x y 
3 n
7.  5 3n 2 
x y z 
2
 3  9
=  2 2n 2  = 4 4n 4
x y z  x y z

9. −3 3 3
 4  n n
8.   = =
 4  64
n
−3 −3
 3x y   x 
3b 2b
9.  b 3 =  2
 −6x y   −2y 
2 3
 −2y  −8y 6
=  2b  = 6b
 x  x