Anatomy of exponential terms

We use exponential notation to write repeated multiplication. For example [latex]10\cdot10\cdot10[/latex] can be written more succinctly as [latex]10^{3}[/latex]. The 10 in [latex]10^{3}[/latex] is called the base. The 3 in [latex]10^{3}[/latex] is called the exponent. The expression [latex]10^{3}[/latex] is called the exponential expression. Knowing the names for the parts of an exponential expression will help you learn how to perform mathematical operations on them.

[latex]\text{base}\rightarrow10^{3\leftarrow\text{exponent}}[/latex]

[latex]10^{3}[/latex] is read as “10 to the third power” or “10 cubed.” It means [latex]10\cdot10\cdot10[/latex], which is 1,000.

[latex]8^{2}[/latex] is read as “8 to the second power” or “8 squared.” It means [latex]8\cdot8[/latex], or 64.

[latex]5^{4}[/latex] is read as “5 to the fourth power.” It means [latex]5\cdot5\cdot5\cdot5[/latex], or 625.

[latex]b^{5}[/latex] is read as “b to the fifth power.” It means [latex]{b}\cdot{b}\cdot{b}\cdot{b}\cdot{b}[/latex]. Its value will depend on the value of b.

The exponent applies only to the number or variable that it is attached to. Therefore, in the expression [latex]xy^{4}[/latex], only the y is affected by the 4. [latex]xy^{4}[/latex] means [latex]{x}\cdot{y}\cdot{y}\cdot{y}\cdot{y}[/latex]. The x is not being raised to the 4th power.

Consider an expression with a negative, like [latex]−3^{4}[/latex].  For reasons related to our discussion in the previous paragraph, only the 3 is being raised to the 4th power, not the negative.  You may find it helpful to even rewrite the expression as [latex]-1\cdot 3^4[/latex].  We would compute this as [latex]–\left(3\cdot3\cdot3\cdot3\right)[/latex] or [latex]−81[/latex].

If [latex]−3[/latex] is to be the base, it must be written using parentheses as [latex]\left(−3\right)^{4}[/latex], which means [latex]−3\cdot−3\cdot−3\cdot−3[/latex], or 81.

Likewise, [latex]\left(−x\right)^{4}\hspace{.02in}=\hspace{.02in}\left(−x\right)\cdot\left(−x\right)\cdot\left(−x\right)\cdot\left(−x\right)=x^{4}[/latex], while [latex]−x^{4}\hspace{.02in}=\hspace{.02in}–\left(x\cdot x\cdot x\cdot x\right)[/latex].

You can see that there is quite a difference, so you have to be very careful! The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication.

In the following video you are provided more examples of applying exponents to various bases.

Caution! Whether to include a negative sign as part of a base or not often leads to confusion. To clarify whether a negative sign is applied before or after the exponent, here is an example.

 

What is the difference in the way you would evaluate these two terms?

1. [latex]-{3}^{2}[/latex]
2. [latex]{\left(-3\right)}^{2}[/latex]

To evaluate 1), you would apply the exponent to the three first, then apply the negative sign last, like this:

[latex]\begin{array}{c}-\left({3}^{2}\right)\\=-\left(9\right) = -9\end{array}[/latex]

To evaluate 2), you would apply the exponent to the 3 and the negative sign:

[latex]\begin{array}{c}{\left(-3\right)}^{2}\\=\left(-3\right)\cdot\left(-3\right)\\={ 9}\end{array}[/latex]

The key to remembering this is to follow the order of operations. The first expression does not include parentheses so you would apply the exponent to the integer 3 first, then apply the negative sign. The second expression includes parentheses, so hopefully you will remember that the negative sign also gets squared.

In the next sections, you will learn how to simplify expressions that contain exponents. Come back to this page if you forget how to apply the order of operations to a term with exponents, or forget which is the base and which is the exponent!

Apply the Product Rule to simplify an expression

Exponential notation was developed to write repeated multiplication more efficiently. There are times when it is easier or faster to leave the expressions in exponential notation when multiplying or dividing. Let’s look at rules that will allow you to do this.

For example, the notation [latex]5^{4}[/latex] can be expanded and written as [latex]5\cdot5\cdot5\cdot5[/latex], or 625. And don’t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.

What happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[/latex]. Expanding each exponent, this can be rewritten as [latex]\left(2\cdot2\cdot2\right)\left(2\cdot2\cdot2\cdot2\right)[/latex] or [latex]2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2[/latex]. In exponential form, you would write the product as [latex]2^{7}[/latex]. Notice that 7 is the sum of the original two exponents, 3 and 4.

What about [latex]{x}^{2}{x}^{6}[/latex]? This can be written as [latex]\left(x\cdot{x}\right)\left(x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\right)=x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}[/latex] or [latex]x^{8}[/latex]. And, once again, 8 is the sum of the original two exponents. This concept can be generalized in the following way:

When multiplying more complicated terms, multiply the coefficients and then multiply the variables.

Apply the Quotient Rule to simplify an expression

Let’s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.

[latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}[/latex]

You can rewrite the expression as: [latex] \displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}[/latex]. Then you can divide off the common factors of 4 in the numerator and denominator: [latex] \displaystyle [/latex]

Finally, this expression can be rewritten as [latex]4^{3}[/latex] using exponential notation. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2.

So, [latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[/latex].

Be careful that you subtract the exponent in the denominator from the exponent in the numerator.

So, to divide two exponential terms with the same base, subtract the exponents.

When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.

In the following video we show another example of how to use the quotient rule to divide exponential expressions

Apply the Power Rule to simplify an expression

Another word for exponent is power.  You have likely seen or heard an example such as [latex]3^5[/latex] can be described as 3 raised to the 5th power. In this section we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent.  We will also learn what to do when numbers or variables that are divided are raised to a power.  We will begin by raising powers to powers.

Let’s simplify [latex]\left(5^{2}\right)^{4}[/latex]. In this case, the base is [latex]5^2[/latex] and the exponent is 4, so you multiply [latex]5^{2}[/latex] four times: [latex]\left(5^{2}\right)^{4}=5^{2}\cdot5^{2}\cdot5^{2}\cdot5^{2}=5^{8}[/latex] (using the Product Rule—add the exponents).

[latex]\left(5^{2}\right)^{4}[/latex] has two powers; it indicates that 5 to the second power is all being raised to the fourth power. And we saw above that the answer is [latex]5^{8}[/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\cdot4=8[/latex].

So, [latex]\left(5^{2}\right)^{4}=5^{2\cdot4}=5^{8}[/latex] (which equals 390,625, if you do the multiplication).

Likewise, [latex]\left(x^{4}\right)^{3}=x^{4\cdot3}=x^{12}[/latex]

This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\left(2^{3}\right)^{5}=2^{15}[/latex].

Raise a product to a power to simplify an expression

Simplify this expression.

[latex]\left(2a\right)^{4}=\left(2a\right)\left(2a\right)\left(2a\right)\left(2a\right)=\left(2\cdot2\cdot2\cdot2\right)\left(a\cdot{a}\cdot{a}\cdot{a}\right)=\left(2^{4}\right)\left(a^{4}\right)=16a^{4}[/latex]

Notice that the exponent is applied to each factor of 2a. So, we can eliminate the middle steps.

[latex]\begin{array}{l}\left(2a\right)^{4} = \left(2^{4}\right)\left(a^{4}\right)\text{, applying the }4\text{ to each factor, }2\text{ and }a\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,=16a^{4}\end{array}[/latex]

The product of two or more numbers/variables raised to a power is equal to the product of each number/variable raised to the same power.

If the variable has an exponent with it, use the Power Rule: multiply the exponents.

Raise a quotient to a power to simplify an expression

Now let’s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex] \displaystyle \frac{3}{4}[/latex] and raise it to the 3^rd power.

[latex] \displaystyle {{\left( \frac{3}{4} \right)}^{3}}=\left( \frac{3}{4} \right)\left( \frac{3}{4} \right)\left( \frac{3}{4} \right)=\frac{3\cdot 3\cdot 3}{4\cdot 4\cdot 4}=\frac{{{3}^{3}}}{{{4}^{3}}}[/latex]

You can see that raising the quotient to the power of 3 can also be written as the numerator, 3, raised to the power of 3, and the denominator, 4, raised to the power of 3.

Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to the power.

[latex] \displaystyle {{\left( \frac{a}{b} \right)}^{4}}=\left( \frac{a}{b} \right)\left( \frac{a}{b} \right)\left( \frac{a}{b} \right)\left( \frac{a}{b} \right)=\frac{a\cdot a\cdot a\cdot a}{b\cdot b\cdot b\cdot b}=\frac{{{a}^{4}}}{{{b}^{4}}}[/latex]

When a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below.

[latex] \displaystyle {{\left( \frac{a}{b} \right)}^{4}}=\frac{{{a}^{4}}}{{{b}^{4}}}[/latex]

This rule is often combined earlier rules we have seen, as demonstrated in the following example.

In the following video you will be shown more examples of simplifying quotients that are raised to a power.

Define and use the Zero Exponent Rule

When we defined the quotient rule, we only worked with expressions like the following: [latex]\displaystyle\frac{{{4}^{9}}}{{{4}^{4}}}[/latex], where the exponent in the numerator (up) was greater than the one in the denominator (down), so the final exponent after simplifying was always a positive number, and greater than zero. In this section, we will explore what happens when we apply the quotient rule for exponents and get a negative or zero exponent.

What if the exponent is zero?

To see how this is defined, let us begin with an example. We will use the idea that dividing any number by itself gives a result of 1. Note, we must assume [latex]x\neq 0[/latex] in the example to avoid division by 0.

[latex]\displaystyle\frac{x^{n}}{x^{n}}=\frac{\cancel{x^{n}}}{\cancel{x^{n}}}=1[/latex]

If we were to instead simplify the original expression using the quotient rule, we would have

[latex]\displaystyle\frac{{x}^{n}}{{x}^{n}}={x}^{n - n}={x}^{0}[/latex]

If we equate the two answers, the result is [latex]{x}^{0}=1[/latex]. Since [latex]x[/latex] can be any nonzero number, it follows that for any nonzero real number [latex]a[/latex],

The sole exception is the expression [latex]{0}^{0}[/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined, or DNE (Does Not Exist).

As done previously, to evaluate expressions containing exponents of 0 or 1, substitute the value of the variable into the expression and simplify.

In the following video there is an example of evaluating an expression with an exponent of zero, as well as simplifying when you get a result of a zero exponent.

Define and use the Negative Exponent Rule

We proposed another question at the beginning of this section.  Given a quotient like [latex] \displaystyle \frac{{{2}^{m}}}{{{2}^{n}}}[/latex] what happens when n is larger than m? We will need to use the negative rule of exponents to simplify the expression so that it is easier to understand.

Let’s look at an example to clarify this idea. Given the expression:

[latex]\displaystyle\frac{{h}^{3}}{{h}^{5}}[/latex]

Expand the numerator and denominator, all the terms in the numerator will divide to 1, leaving two hs multiplied in the denominator, and a numerator of 1.

We could have also applied the quotient rule from the last section, to obtain the following result:

[latex]\displaystyle\begin{array}{r}\frac{h^{3}}{h^{5}}\,\,\,=\,\,\,h^{3-5}\\\\=\,\,\,h^{-2}\,\,\end{array}[/latex]

Putting the answers together, we have [latex]\displaystyle{h}^{-2}=\frac{1}{{h}^{2}}[/latex]. This is true when h, or any variable, is a real number and is not zero.

Let’s looks at some examples of how this rule applies under different circumstances.

In the following video you will see examples of simplifying expressions with negative exponents.

What if we need to apply multiple exponent rules to one problem? That will be our focus in the next section, but try the Think About It problem below to explore this idea.

Summary

• Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.
• The product rule for exponents: For any number x and any integers a and b, [latex]\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}[/latex].
• The quotient rule for exponents: For any non-zero number x and any integers a and b: [latex] \displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[/latex]
• The power rule for exponents:
  1. For any nonzero numbers a and b and any integer x, [latex]\left(ab\right)^{x}=a^{x}\cdot{b^{x}}[/latex].
  2. For any number a, any non-zero number b, and any integer x, [latex] \displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}[/latex]