Multiplying and Dividing 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.

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:

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)(x²y • x²y) = 12x²⁺²y¹⁺¹ = 12x⁴y²
    
   
2. 3ab • 5xy² = (3 • 5)abxy² = 15abxy²

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.

Divide the following terms.

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

Divide the following terms.

1.  b⁶ ¸ b³ = b⁶/b³ = b^6–3 = b³
    
   
2. k⁴z⁹/ k²z⁵ = k^4–2z^9–5 = k²z⁴
    
   
3. 20a⁴b⁵ / 5a⁷b⁴ = (20 ÷ 5)a^4–7b^5–4 = 4a^–3b
   This solution can be rewritten as 4a^–3b = 4b/a³

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

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