Simplifying Multiplication Lessons
Using methods from arithmetic, you can multiply two numbers like,
(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:
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,
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,
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
x^{2} · 22 · x^{2}
x^{2}(2)
(x^{2})(2)
x^{2} * 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 x^{3}, 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
Multiplying Variables
The first problem we will work on is
When two of the same variables are multiplied (in this case both x) the answer consists of the varable with an exponent that is the sum of the other two exponents.
In this case, both terms in the multiplication are x without a superscripted exponent, therefore each has an implied exponent of 1. We are multiplying so we will add the exponents, 1 + 1, to get 2. Therefore, the answer is x to the second power:
The next problem is
This time we are multiplying x and x to the fourth power. The first multiplier, x, doesn't have a visible exponent. As before we know that x has an implied exponent of 1. The second multiplier, x^{4}, has a visible exponent, 4.
Both multipliers are the same letter, so we multiply by adding the exponents. Since 1 + 4 = 5, the answer is:
Multiplying Variables
More Than One Variable
In a problem like the one below, the terms cannot be multiplied by simply adding exponents because each multiplier is a different letter variable.
To show the answer to this problem, you simply remove the multiply sign and put the terms next to each other, and in alphabetical order:
Look over the next problem.
As you can see, in this problem the x^{2} is multiplying a term which has a variable x, and a variable y. The result of this multiplication is found using a combination of the two previously presented techniques.
We begin by adding the exponents of x^{2} and x^{3}: 2 + 3 = 5, therefore the product will contain x^{5}.
Now since a y is present in the second term but not the first, it is simply copied into the result. Thus the answer is:
Multiplication Resources
Multiplying Practice Problems / Worksheet Multiplication Calculator
Next Lesson:
Distributive Property |
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