Textbook

In this chapter we will explore what exponents mean as well as how we can use them in expressions and algebraic equations.

Exponents are the small numbers that go on the upper right diagonal of a number or variable ($2_{4},x_{2},10_{−3}$). The way exponents work is that we start with $1$ and multiply it by our number as many times as the exponent tells us to. We also call this **putting a number to a “power.”** For instance, let’s do an example of $2_{4}$:

Start with $1$, then multiply it by our number ($2$) a total of four times (the exponent):

$2_{4}=1∗2∗2∗2∗2=16$

See below for a more detailed breakdown:

(spoiler)

$1∗22∗24∗28∗2 =2=4=8=16 $

$10_{4}$ has an exponent of $4$, so we call it “ten to the fourth power.” We will multiply $1$ by our number ($10$) four times: $10∗10∗10∗10=10,000$

$3_{2}$ has an exponent of $2$, so we call it “three to the second power.” We will multiply $1$ by our number ($3$) two times: $3∗3=9$. **A number to the second power is often referred to as “squared.”** In this case, it is more accurate to call our number “three squared.”

Negative exponents are very similar to positive exponents in that they indicate how many times we will use our number. The difference is that instead of **multiplying** $1$ by our number an amount of times equal to the exponent, we will **divide** $1$ by our number an amount of times equal to our exponent. Let’s use a similar example as above, $2_{−4}$:

Start with $1$, then divide it by our number ($2$) a total of four times (the exponent):

$1/(2∗2∗2∗2)=1/16=0.0625$

See below for a more detailed breakdown:

(spoiler)

$1/2(1/2)/2(1/4)/2(1/8)/2 =1/2=1/4=1/8=1/16 $

A shortcut way to understanding this is flipping which side of the division bar the number is on. For example:

$2_{−2}=1/(2_{2})=1/4$

Or:

$1/(2_{−2})=2_{2}=4$

You can think about a negative exponent as a way to move anything from the numerator to the denominator and vice versa.

There are two different situations involving exponents that we will mention here. The first, is when the inside of the parentheses does not include addition/subtraction. The second is when the inside does include addition/subtraction.

For the first situation, when there is no addition/subtraction within the parentheses, we will apply the exponent to each number or variable inside. In the following example, you will see that the exponent, $2$, is applied to all the numbers and variables within the parentheses:

$(3x)_{2}=3_{2}∗x_{2}$

With our second situation, when there is addition/subtraction within the parentheses, we will have to simplify the exponent in a special way: you have to FOIL the expression in the parentheses. If you are unfamiliar with the concept of FOIL, see the chapter Simplifying Expressions

To visualize the need for this a little bit better, you can write out the exponent as follows"

$(x+2)_{2}=(x+2)(x+2)$

This is because the exponent has the expression multiplied by itself. You will have to FOIL it because of this. So, the simplified form of this expression would be

$(x+2)(x+2)=x_{2}+4x+4$

When you have a fraction in which the **whole fraction** is raised to an exponent, both the top and bottom of the fraction must be put to that exponent

$((x+1)2x )_{2}=(x+1)_{2}(2x)_{2} $

After this is done, then carry out the rest of the simplification according to the rules discussed in the chapter “Rules of simplification”. To finish the above example of exponents, we will distribute the exponent on top and FOIL the denominator:

$(x_{2}+2x+1)4x_{2} $

When simplifying expressions, we have to follow the order of operations: PEMDAS. Exponents are the second thing we should look for in the order of operations. Your question should be “What do we do with them?” You will want to simplify exponents just as we have done above. This way, we can use the number more easily with the rest of the expression.

Try an example. When you get to the point in your order of operations, simplify the number with the exponent before continuing to solve the expression:

Simplify the following expression:

$2+3(4_{2}−2(3))$

(spoiler)

Parentheses: Start with the innermost parentheses $2(3)=6$. Then, we need to work on the next step within the parentheses. However, the next step involves an exponent, so we must simplify this exponent first.

Exponents: $4_{2}=1∗4∗4=16$. Now we can finish the step inside the parentheses $16−6=10$.

Multiplication: Multiply the final number within the parentheses with the number outside the parentheses $3∗10=30$.

Addition: $2+30=32$

So, our simplified expression is $32!$

When multiplying variables with exponents, you will first multiply any numbers that come before the two numbers as you normally would. Then, you will take the *exponents* of the variables you are multiplying, and add them together. The sum of the two will be the exponent of the new term. **Remember that if a variable ($x$) has no number in the exponent, it is an exponent of $1$.**

$x_{2}∗x_{3}=x_{2+3}=x_{5}$

Division works in a similar way. If there are numbers in front of the variables, then you will divide them first. Then, the new exponent will be the difference between the first and second exponents.

$x_{2}x_{3} =x_{3−2}=x_{1}=x$

Try some examples:

What is $x∗x_{3}$ simplified?

(spoiler)

$x∗x_{3}=x_{1+3}=x_{4}$

What is $2x_{2}∗3x_{3}$ simplified?

(spoiler)

First, multiply the numbers in front:

$2∗3=6$

Then multiply the variables by adding the exponents together:

$x_{2}∗x_{3}=x_{2+3}=x_{5}$

Now put the two multiplied values together for the final, simplified answer!

$6x_{5}$

What is $4x_{2}/2x$ simplified?

(spoiler)

First, divide the numbers in front:

$4/2=2$

Then divide the variables by subtracting the second exponent from the first:

$x_{2}/x=x_{2−1}=x$

Now put the two divided values together for the final, simplified answer!

$2x$

A number may have an exponent raised to another exponent, which looks like this:

$(x_{2})_{2}$

When a number like this comes up, you should start from the furthest exponent and work down to the number or variable. In this example, we would first evaluate the exponents, $2_{2}$, which gives us $4$, so we can simplify the value of the above expression to $x_{4}$. Performing this simplification will also help you see how to use it in equations. Simplify it first!

Another way to think about it is to think of the two exponents being multiplied together. This is important because sometimes it’s not always a number that’s in the exponent; sometimes it’s a variable. For example, if we have $(2_{3x})_{2y}$, we can multiply the “power to a power” together to create a combined exponent, like this: $2_{3x∗2y}=2_{6xy}$

Perhaps you have seen an equation that looks like this:

$x_{3}+x_{2}+x+1=0$

The key is that this equation is **already in its simplified form** because you may **only add or subtract numbers with the same exponents**.

This is to say that you may not simplify $x_{3}+x_{2}$ any more than it already appears. You have to treat any $x$ with a different exponent as if it were its own separate variable. However, you could simplify if they were both of the same exponents:

$x_{3}+2x_{3}=3x_{3}$

You have automatically always followed this rule without knowing it by adding numbers with the exponent of $1$:

$4+6−3=4_{1}+6_{1}−3_{1}=7$

For details on how to reverse exponents in algebra, i.e., how to solve for $x$ when $x$ has an exponent, see the chapter Radicals and rational exponents.

Sign up for free to take 40 quiz questions on this topic

All rights reserved ©2016 - 2024 Achievable, Inc.