When multiplying algebraic terms you (1) multiply numerical
coefficients together, then
(2) list all the variables that occur in the terms being multiplied
and (3) add the exponents of like variables. When doing this,
we can use what we learned in the last section on exponents. Below
are some examples of multiplication applied to algebraic terms.
In this example, the base of both terms is x. This means we only need to multiply the coefficient for each term, attach the variable, then add the exponents. Notice that when no exponent is explicitly stated, the exponent is assumed to be one. 
x^{2} • 3x = 1 • 3 • x^{2+1 }= 3x^{3} 
Here, we multiply the numerical coefficients, keep the variable and add the exponents of like variables. 
5k^{3} • k^{2} = 5 • 1 • k^{3+2} = 5k^{5} 
Both examples above contained terms with the same variables in their bases. However, even when the bases are not the same, we follow the same procedure. We multiply the numerical coefficients and then simply list the variables from each term. For example, if we multiply the terms 3xy and 5y^{2}, we get the result 15xy^{3}. This works because we can view 5y^{2 }as 5x^{0}y^{2}. (NOTE: In the previous section we mentioned that x^{0} = 1.) So we can write this multiplication as:
Multiply the following terms.
First multiply the numerical coefficients, then use the exponent rules.
The process for division of algebraic expressions is much like the multiplication process, with the exception that you divide the numerical coefficients and then subtract exponents instead of adding. The exponent rules for division are:

If we divide 12xy^{2} by 3y we get 4xy as a result. This works because we can look at the term in the bottom of the fraction as being 3x^{0}y^{1}.
This subtraction of exponents looks as if we are canceling out numbers and variables. This is often expressed in the way shown below. (If you feel you need to review how to work with fractions, you should return to Book I of this series, Review of Fractions.)
Because we subtract exponents, it is possible to get a negative exponent. NOTE: When we get an answer in this form, with negative exponents, we may leave them in this form, or rewrite the expression according to the following rule.
So if we have divide 2kr by kr^{4} we get:
Now let's look at an example of division problems.
Divide the following terms.
Divide the following terms.
Below is a brief outline of the steps for multiplying and dividing terms that contain variables with exponents.
6 z^{a + c}x^{d + b}
3x^{c}^{–1}y^{a}^{–b} 
[introduction]  [exponents]  [multiply/divide]  [add/subtract]  [grouping] 