# Multiplying and Dividing Algebraic Terms

## Multiplying Algebraic Terms

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. x2 • 3x = 1 • 3 • x2+1 = 3x3 Here, we multiply the numerical coefficients, keep the variable and add the exponents of like variables. 5k3 • k2 = 5 • 1 • k3+2 = 5k5

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 5y2, we get the result 15xy3. This works because we can view 5y2 as 5x0y2. (NOTE: In the previous section we mentioned that x0 = 1.) So we can write this multiplication as:

3xy · 5y2 = 3x1y1 · 5x0y2 = 3 · 5 · x1+0 · y1+2 = 15xy3

### Example

Multiply the following terms.

1. 2x2y • 6x2y
2. 3ab • 5xy2

First multiply the numerical coefficients, then use the exponent rules.

1. 2x2y • 6x2y = (2 • 6)(x2yx2y) = 12x2+2y1+1 = 12x4y2

2. 3ab • 5xy2 = (3 • 5)abxy2 = 15abxy2

## Dividing Algebraic Terms

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:

 Exponent Rules for Division If there are numerical coefficients in the expressions to be divided, just divide the numerical coefficient and then use the exponent rule to divide the variables. To divide variables, keep the variables and subtract the exponents.

If we divide 12xy2 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 3x0y1.

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.

x–n = 1/xn

So if we have divide 2kr by kr4 we get:

Now let's look at an example of division problems.

### Example

Divide the following terms.

1. b6 ¸ b3
2. k4z9/k2z 5
3. 20a4b5 / 5a7b4

Divide the following terms.

1.  b6 ¸ b3 = b6/b3 = b6–3 = b3

2. k4z9/ k2z5 = k4–2z9–5 = k2z4

3. 20a4b5 / 5a7b4 = (20 ÷ 5)a4–7b5–4 = 4a–3b
This solution can be rewritten as 4a–3b = 4b/a3

## Multiplying and Dividing Algebraic Terms

Below is a brief outline of the steps for multiplying and dividing terms that contain variables with exponents.

 Multiplying Terms Multiply the numerical coefficients. (Do not confuse the numerical coefficients with the base.) Perform the multiplication on the variables: Keep the variables from each term. For each variable, add the exponents that appear for each variable. 2zaxd • 3zcxb = (2 • 3)za + cxd + b + b = 6 za + cxd + b   Dividing Terms Divide the numerical coefficients. (Do not confuse the numerical coefficients with the base.) Perform the division on the variables: Keep the variables from each term. For each variable, subtract the exponent on the bottom of the fraction from the exponent on the top of the fraction. 6xcya/2xyb = (6/2)xc–1ya–b = 3xc–1ya–b