Properties of exponents

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

Understanding exponents

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):

$2^4=1\ast 2\ast 2\ast 2\ast 2=16$

See below for a more detailed breakdown:

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

$3^2$ has an exponent of $2$, so we call it “three to the second power.” We will multiply $1$ by our number ($3$) two times: $3\ast 3=9$. 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.”

Understanding negative exponents

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):

$$1/(2\ast 2\ast 2\ast 2)=1/16=0.0625$$

See below for a more detailed breakdown:

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.

Exponents on parentheses

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:

$$(3x{)}^2=3^2\ast x^2$$

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

$$(x+2)(x+2)=x^2+4x+4$$

Exponents on fractions

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

$$(\frac{2x}{(x+1)}{)}^2=\frac{(2x{)}^2}{(x+1{)}^2}$$

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:

$$\frac{4x^2}{(x^2+2x+1)}$$

Exponents in arithmetic

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:

$$2+3(4^2-2(3))$$

Exponents in algebra

Multiplying and dividing with exponents

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 $1$.

$$x^2\ast x^3=x^{2+3}=x^5$$

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.

$$\frac{x^3}{x^2}=x^{3-2}=x^1=x$$

Try some examples:

What is $x\ast x^3$ simplified?

What is $2x^2\ast 3x^3$ simplified?

What is $4x^2/2x$ simplified?

Exponent raised to an exponent (double exponents)

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: $2^{3x\ast 2y}=2^{6xy}$

Adding and subtracting with exponents

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 $1$:

$$4+6-3=4^1+6^1-3^1=7$$

Reversing exponents

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.

Key points

Power. A number “to a power of…” means that it has that “power” as an exponent. This exponent indicates how many times we multiply $1$ by our original number.

Square. A number is squared when it is to the second power (exponent of $2$).

Multiplying/dividing. Multiply or divide any numbers in front of the variables, then multiply or divide the variables. This is when the exponent rule will come into play: add exponents when multiplying, and subtract them when dividing.

Adding/subtracting. Add or subtract variables.

Double exponents. Think of PEMDAS, and take care of the exponent that is deepest into the term first. Find the value of the exponent raised to another exponent, and that will be the new exponent of the actual number.