Exponents

In this lesson we will look at the definition of an exponent and how to multiply together two expressions that have exponents.

We will introduce the basic definition using an example.  Consider the expression

We see that this is a multiplication problem where 2 is multiplied by itself 5 times.  Just like repeated addition becomes multiplication, there is a mathematical operation, called exponentiation, that represents repeated multiplication.  We can write the expression above as

Here the number 2 which represents the number that we are multiplying is called the base and the number 5 which represents the number of times the base is multiplied by itself is called the exponent.

Example 1

Expand x⁴ as a repeated multiplication problem.

Solution

We see the base is "x" and the exponent is 4.  Using the definition, we can write x multiplied by itself 4 times.  We have

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 1

Expand w⁶ as a repeated multiplication problem

Answer

Sometimes the base is more complicated, but if we carefully follow the definition of an exponent, we can expand the exrpession as a multiplication problem.  The next example demonstrates this.

Example 2

Expand and simplify

        (-2x)³

Solution

First recognize the base as (-2x) and the exponent as 3.  Therefore (-2x) is multiplied by itself 3 times.  The use of parentheses here will help us avoid the mistake of confusing the negative sign and the subtraction sign.  In this problem subtraction does not occur.  We write

        (-2x)³  =  (-2x)(-2x)(-2x)

Now to simplify.  First notice that there are 3 negative signs.  Since 3 is an odd number, the result will be negative.  (multiplying an odd number of negative numbers results in a negative number and and multiplying an even number of negative numbers results in a positive number.  We can write

        =  -(2x)(2x)(2x)

Next we can rearrange the terms so that the numbers are together and the x's are together.  We get

        =  -(2)(2)(2)(x)(x)(x)

Now, 2 times 2 times 2 is 8 so we can write

        =  - 8x(x)(x)

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 2

Expand (-3y)⁴ as a repeated multiplication problem

Answer

In each of the previous examples and exercises, the base involved a variable.  As we know, variables represent numbers.  Sometimes an exponential expression is given where the base involves a variable and we are asked what is the value of the expression when the variable is a given number.  The following example illustrates this.

Example 3

Evaluate the expression x⁵ when x is -2.

Solution

First let's write x⁵ in expanded form:

Now we can substitute -2 for x in this expression.  It is always a good idea to wrap parentheses around what we are substituting.  We have

Now we can count the number of negative signs.  There are 5 negative signs, so the final answer will be negative.  Now we calculate

        -(2)(2)(2)(2)(2)   =  -(4)(2)(2)(2)

                                 =  -(8)(2)(2)

                                 =  -(16)(2)

                                 =   -32

So the final answer is -32.

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 3

Evaluate (2x)³ when x  =  -3.

Answer

Now that we understand how to read exponents, we are ready to see how to multiply two expression that involve exponents.  We consider an example

Example 4

Multiply

Solution

We begin by expanding:

Notice that the right hand side is a multiplication problem with x multiplied by itself 4 times and then again 2 times.  The total number of x's that appear in this expression is

        4 + 2  =  6

The general rule is

Whenever we are multiplying such expressions with the same base, we always add the exponents.

We call this the addition rule for exponents.  We can conclude that

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 4

Multiply (k⁹ )(k¹¹)

Answer

Sometimes we will be asked to use both the addition rule for exponents and the idea of rearrangement.  The next examples will make use of this combination.

Example 5

Multiply and simplify

Solution

We first rearrange the factors so that the leading numbers, called the coefficients, are grouped together and the variables are grouped together.  We also write x  as x¹ so  that we can realize it as an exponent.

For the coefficients, we have

        (2)(5)  =  10

We use the addition rule for the exponents with bases x and y.

        2 + 1  =  3        and        3 + 4  =  7

We can put this all together to get an answer of

        10x³y⁷z⁷

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 5

Multiply (3a⁴bc³ )(4ab⁵)

Answer