Simplifying Multiplication

Using methods from arithmetic, you can multiply two numbers like,

5 * 11

(Note * is used as the multiplication symbol.) When solving problems using algebra, it is often important to be able to multiply algebraic terms which have variables. Examine this problem:

5x * 11x2

Clearly the methods for multiplying numbers in arithmetic do not apply here. This lesson explains a method for doing this multiplication.

Before actually walking through such multiplication problems, we will present a few concepts which are important to understand before learning the multiplication technique.

Commutative Property

The Commutative Property of Multiplication is a statement or observation about multiplication which indicates that the product of a multiplication problem is the same, regardless of the order the terms were multiplied in. For example,

2 * 10 = 20

and

10 * 2 = 20

Note that the product of 2 and 10 is 20 and that the product of 10 and 2 is 20, the products are the same. It turns out that this is true for real numbers in general. That is, the order in which two numbers are multiplied does not affect the result.

This property can be extended to the case where we multiply more than 2 terms. For example,

4 * 6 * 5 = 120

and

6 * 5 * 4 = 120

Notation

The next important concept is understanding the various styles of writing multiplication in algebra. So far we have been using the * symbol in this lesson to denote multiplication. There are several other mothods of showing multiplication which are also acceptable. Consider the examples below. Each line shows one way of writing "x squared times two".

Various Multiplication Notation Examples

x2 · 2

2 · x2

x2(2)

(x2)(2)

x2 * 2

You may recall using a "x" as the multiplication symbol in arithmetic. This is generally avoided in algebra because x is the most common variable used in expressions and equations. It would be extremely confusing if x were used for both a variable and a multiplication sign.

In this lesson, we will continue to use * to represent multiplication because it is easily entered with the keyboard, and because this notation is consistent with our calculators and many handheld calculators.

Implied Exponents

Finally, it is important to understand the implied exponent on some variables. When we consider a term like x3, we know that x has an exponent of three. But what about the term x?

When a variable does not have a superscript which indicates its exponent, it has an implied exponent of 1. Thus, x has an exponent of 1 and we can say that

x1 = x