Learning Objectives
- Interpret the meaning of integer exponents.
- Explain the derivation of the product and power rules for exponents.
- Correctly execute calculations involving the product and power rules for exponents.
Chapter 6 Exponents
Section 6.1 Don’t Memorize These Formulas
In an earlier section, we had worked with monomial and polynomial expressions that involved exponents. We implicitly assumed that you were familiar with the notation. We’re going to formally define exponents here for the sake of completeness.
Definition 6.1. Exponent.
A positive integer exponent represents repeated multiplication. The expression \(a^n\) represents the quantity \(a\) multiplied by itself \(n\) times. In other words,
\begin{equation*}
a^n = \underbrace{a \cdot a \cdots a}_{\text{$n$ times}}.
\end{equation*}
This definition allows us to write out exponent expressions "the long way" by explicitly writing out the terms in the product:
\begin{equation*}
\begin{aligned}
x^1 \amp = x \\
x^2 \amp = x \cdot x \\
x^3 \amp = x \cdot x \cdot x \\
\amp \vdots
\end{aligned}
\end{equation*}
Activity 6.1. The Product Rule for Exponents.
This definition has a consequence that can be seen through some examples. Consider the following products:
\begin{equation*}
\begin{aligned}
x^3 \cdot x^4 =
\underbrace{\underbrace{(x \cdot x \cdot x)}_{\text{3 times}} \cdot
\underbrace{(x \cdot x \cdot x \cdot x)}_{\text{4 times}}}_{\text{7 times}} =
x^7 \\ \\
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
x^4 \cdot x^1 =
\underbrace{\underbrace{(x \cdot x \cdot x \cdot x)}_{\text{4 times}} \cdot
\underbrace{(x)}_{\text{1 time}}}_{\text{5 times}} =
x^5
\end{aligned}
\end{equation*}
Try it!
Based on the pattern observed above, what would you say \(x^m \cdot x^n\) should be equal to? Write out an explanation in both words and an equation (similar to the ones above) that explains why the pattern exists.
Solution.
\begin{equation*}
\begin{aligned}
x^m \cdot x^n \amp = \underbrace{\underbrace{(x \cdot x \cdots x)}_{\text{$m$ times}} \cdot \underbrace{(x \cdot x \cdots x)}_{\text{$n$ times}}}_{\text{$m + n$ times}} = x^{m + n}
\end{aligned}
\end{equation*}
Since \(x^m\) means to multiply \(x\) by itself \(m\) times, and \(x^n\) means to multiply \(x\) by itself \(n\) times, if you do the first then the second, you’ve multiplied \(x\) by itself \(m + n\) times in total.
Activity 6.2. The Power Rule for Exponents.
There is a second consequence of the definition of exponents that can also been seen through some examples:
\begin{equation*}
\begin{aligned}
\left( x^4 \right)^2 =
\underbrace{ \underbrace{(x \cdot x \cdot x \cdot x)}_{\text{4 times}} \cdot
\underbrace{(x \cdot x \cdot x \cdot x)}_{\text{4 times}} }_{\text{2 groups of 4 times}} =
x^8 \\ \\
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\left( x^2 \right)^3 =
\underbrace{\underbrace{(x \cdot x)}_{\text{2 times}} \cdot
\underbrace{(x \cdot x)}_{\text{2 times}} \cdot
\underbrace{(x \cdot x)}_{\text{2 times}}}_{\text{3 groups of 2 times}} =
x^6
\end{aligned}
\end{equation*}
Try it!
Based on the pattern observed above, what would you say \(\left( x^m \right)^n\) should be equal to? Write out an explanation in both words and an equation (similar to the ones above) that explains why the pattern exists.
Solution.
\begin{equation*}
\begin{aligned}
\left( x^m \right)^n \amp = \underbrace{ \underbrace{(x \cdot x \cdots x)}_{\text{$m$ times}} \cdot \underbrace{(x \cdot x \cdots x)}_{\text{$m$ times}} \cdots \underbrace{(x \cdot x \cdots x)}_{\text{$m$ times}} }_{\text{$n$ groups of $m$ times}} = x^{mn}
\end{aligned}
\end{equation*}
The product of \(x\) multiplied by itself \(n\) times is multiplied by itself \(m\) times, giving you \(m\) groups of \(x\) multiplied by itself \(n\) times, for a total of \(mn\) times that \(x\) has been multiplied by itself.
These two results are important enough that they have names.
Theorem 6.2. Properties of Exponents.
When \(m\) and \(n\) are positive integers, we have the following:
- The Product Rule for Exponents: \(x^m \cdot x^n = x^{m+n}\)
- The Power Rule for Exponents: \(\left( x^m \right)^n = x^{mn}\)
Did you notice how Definition 6.1 emphasizes that it only applies to positive integers? This is because the definition requires us to count the number of terms in the product. But if that’s the case, what should \(x^0\) be? Should it be 0? Should it be 1? What about negative powers of \(x\text{?}\) Let’s see if we can come up with a sensible pattern. Consider the following diagram:
This gives us an accurate picture of the pattern of exponents. Starting from any equation, we can get the next one by adding 1 to the exponent and multiplying the right side by another \(x\text{.}\) We’re now going to try to turn this around and go backwards:
So all we need to do is continue the pattern.
These ideas lead us to our next definition, which completes the definitions of exponents for the remaining integers:
Definition 6.3. Exponent Notation.
We define the following notation for \(x \neq 0\text{:}\)
- \(x^0 = 1\) (Zero exponent)
- \(x^{-n} = \frac{1}{x^n}\) (Negative exponent)
The condition for this definition is worthy of a closer look. The challenge that arises is that the pattern that we had fails when \(x = 0\text{.}\) The reason is that we cannot divide by zero, so inverting the multiplication and turning it into division simply fails.
There are two other formulas that we can get by combining these properties.
Theorem 6.4. More Properties of Exponents.
When \(x \neq 0\) and \(n\) is any integer:
- \(\displaystyle \left( \frac{1}{x} \right)^{-n} = x^n\)
- \(\displaystyle \frac{1}{x^{-n}} = x^n\)
Activity 6.3. Negative Exponents.
As it turns out, the patterns that were developed above can also be applied when the exponents are not positive integers. Consider the following example:
\begin{equation*}
\begin{aligned}
x^5 \cdot x^{-3} \amp = x^{5 + (-3)} \amp \eqnspacer \amp \text{Product rule for exponents} \\
\amp = x^2 \amp \amp \text{Arithmetic}
\end{aligned}
\end{equation*}
If we apply pattern from above, we get the same result:
\begin{equation*}
\begin{aligned}
x^5 \cdot x^{-3} \amp = x^{5 + (-3)} \amp \eqnspacer \amp \text{Product rule for exponents} \\
\amp = x^2 \amp \amp \text{Arithmetic}
\end{aligned}
\end{equation*}
Try it!
Calculate \(x^2 \cdot x^{-5}\) using both types of presentations above. Your final result should be of the form for some integer \(n\text{.}\)
Solution.
\begin{equation*}
\begin{aligned}
x^2 \cdot x^{-5} \amp = x^2 \cdot \frac{1}{x^5} \amp \amp \text{Definition of negative exponents} \\
\amp = \frac{x^2}{x^5} \amp \amp \text{Multiply fractions} \\
\amp = \frac{x \cdot x}{x \cdot x \cdot x \cdot x \cdot x} \amp \amp \text{Definition of exponents} \\
\amp = \frac{ \cancel{x \cdot x} } {\cancel {x \cdot x} \cdot x \cdot x \cdot x} \amp \amp \text{Reduce the fraction} \\
\amp = \frac{1}{x^3} \amp \amp \text{Simplify} \\
\amp = x^{-3} \amp \amp \text{Definition of negative exponents}
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
x^2 \cdot x^{-5} \amp = x^{2 + (-5)} \amp \amp \text{Product rule for exponents} \\
\amp = x^{-3} \amp \amp \text{Arithmetic}
\end{aligned}
\end{equation*}
Activity 6.4. Properties of Exponents with Negative Exponents.
The power rule for exponents also works when the exponents are zero or negative. You have all the tools you need to demonstrate this for yourself.
Try it!
Calculate \(\left( x^{-2} \right)^3\) using a presentation that shows all of the individual steps. Then verify that the power rule gives the same result.
Solution.
\begin{equation*}
\begin{aligned}
\left( x^{-2} \right)^3 \amp = x^2 \cdot x^2 \cdot x^2 \amp \amp \text{Definition of exponents} \\
\amp = \frac{1}{x^2} \cdot \frac{1}{x^2} \cdot \frac{1}{x^2} \amp \amp \text{Definition of negative exponents} \\
\amp = \frac{1}{x \cdot x} \cdot \frac{1}{x \cdot x} \cdot \frac{1}{x \cdot x} \amp \amp \text{Definition of exponents} \\
\amp = \frac{1}{x \cdot x \cdot x \cdot x \cdot x \cdot x} \amp \amp \text{Multiply fractions} \\
\amp = \frac{1}{x^6} \amp \amp \text{Simplify} \\
\amp = x^{-6} \amp \amp \text{Definition of negative exponents}
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\left( x^{-2} \right)^3 \amp = x^{-2 \cdot 3} \amp \amp \text{Power rule for exponents} \\
\amp = x^{-6} \amp \amp \text{Arithmetic}
\end{aligned}
\end{equation*}
Section 6.2 Worksheets
PDF Version of these Worksheets
1
external/worksheets/06-Worksheets.pdfWorksheet Worksheet 1
1.
Calculate \(t^2 \cdot t^4\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result.
2.
Calculate \(\left( a^3 \right)^4\) using a presentation that shows all of the individual steps. Then verify that the power rule gives the same result.
3.
Calculate \(y^{-3} \cdot y^5\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result.
Worksheet Worksheet 2
1.
Calculate \(x^{-2} \cdot x^{-3}\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result.
2.
Calculate \(\left( x^{-1} \right)^{-n}\) using the power rule. Then rewrite the part of the expression inside the parentheses using the definition of negative exponents. In order for the math to be consistent, the two results should be equal. Explain how this verifies the first formula in Theorem 6.4.
3.
Start from the equation \(x^{-n} = \frac{1}{x^n}\) and take the reciprocal of both sides of the equation. Explain how this verifies the second formula in Theorem 6.4.
Worksheet Worksheet 3
1.
Calculate \(x^{2} \cdot x^{-5}\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result. Give your final answer in the form \(x^n\) for some number \(n\text{.}\)
2.
Calculate \(x^{2n} \cdot x^{3n}\) using the product rule. Explain the logic of your result in complete sentences.
3.
Consider the following presentation:
\begin{equation*}
\begin{aligned}
x^3 \cdot x^{-3} \amp = x^3 \cdot \frac{1}{x^3} \amp \eqnspacer \amp \text{Definition of negative exponents} \\
\amp = \frac{x^3}{x^3} \amp \amp \text{Multiply fractions} \\
\amp = \frac{x \cdot x \cdot x}{x \cdot x \cdot x} \amp \amp \text{Definition of exponents} \\
\amp = \frac{\cancel{x \cdot x \cdot x}}{\cancel{x \cdot x \cdot x}} \amp \amp \text{Reduce the fraction} \\
\amp = 0
\end{aligned}
\end{equation*}
Identify and explain the error. What would you suggest as a way for students to avoid this mistake?
Worksheet Worksheet 4
1.
Calculate \(\left( x^{3} \right)^{-4}\) using a presentation that shows all of the individual steps. Then verify that the power rule gives the same result. Give your final answer in the form \(x^n\) for some number \(n\text{.}\)
2.
Calculate \(\left( x^{2m} \right)^{3n}\) using the power rule. Explain the logic of your result in complete sentences.
3.
Calculate \(x^{4} \cdot x^{-4}\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result.
Worksheet Worksheet 5
1.
Calculate \(\left( x^{-3} \right)^{-4}\) using a presentation that shows all of the individual steps. Then verify that the power rule gives the same result. Give your final answer in the form \(x^n\) for some number \(n\text{.}\)
2.
Calculate \(x^n \cdot x^m \cdot x^p\text{.}\) Explain the logic of your result.
3.
Calculate \(\left( \left( x^n \right)^m \right)^p\text{.}\) Explain the logic of your result.
Section 6.3 Deliberate Practice: Exponents
Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.
Focus on these skills:
- Write the original expression.
- Imagine writing out the various groupings of the variables to reinforce the specific concepts that connect to the formulas.
- Pay close attention to the interplay between negative exponents and fractions.
- Present your work legibly.
Worksheet Worksheet
Instructions: Simplify the expression, expressing your final answer without fractions.
1.
Simplify \(x^3 \cdot x^6 \cdot x^{-5}\text{.}\)
2.
Simplify \(\left( a^{-4} \right)^3 \cdot a^2\text{.}\)
3.
Simplify \(m^3 \cdot m^2 \cdot m^{-5}\text{.}\)
4.
Simplify \(s^2 \cdot \left( s^3 \right)^{-1}\text{.}\)
5.
Simplify \(y^{-3} \cdot y^6 \cdot y^{-4}\text{.}\)
6.
Simplify \((n^3 \cdot n^2)^{-2} \cdot n^4\text{.}\)
7.
Simplify \(\dfrac{b^3 \cdot b^5}{b^5}\text{.}\)
8.
Simplify \(\dfrac{p^4 \cdot p^3}{\left( p^2 \right)^3}\text{.}\)
9.
Simplify \(\dfrac{v^{-2} \cdot \left( v^3 \right)^2}{v^3 \cdot v^{-1}}\text{.}\)
10.
Simplify \(\dfrac{\left( z^{-3} \right)^{-2} \cdot z^4}{z^2 \cdot \left( z^{-3} \right)^{-2}} \text{.}\)
Section 6.4 Closing Ideas
There is a balance between writing out all the steps and just using formulas. Just as with other algebraic steps (such as combining like terms), eventually the expectation is that both the person writing things out and the person reading it will be comfortable enough that all of the individual steps will not need to be written out.
But even as you get more experience, there will come moments when you can’t quite remember the formula. When this happens, there is a mantra you can use to help jog your memory: "When in doubt, write it out!" What this means is that you can always fall back on the basic ideas to recover these formulas.
Remember that exponents, in their most basic form, are just a shorthand notation for repeated multiplication. After that, it’s just logic.
The Product Rule for Exponents:
\begin{equation*}
x^m \cdot x^n
= \underbrace{\underbrace{(x \cdot x \cdots x)}_{\text{$m$ times}} \cdot \underbrace{(x \cdot x \cdots x)}_{\text{$n$ times}}}_{\text{$m +n$ times}}
= x^{m+n}
\end{equation*}
The Power Rule for Exponents:
\begin{equation*}
\left( x^m \right)^n
= \underbrace{ \underbrace{(x \cdot x \cdots x)}_{\text{m times}} \cdot \underbrace{(x \cdot x \cdots x )}_{\text{$m$ times}} \cdots \underbrace{(x \cdot x \cdots x )}_{\text{$m$ times}}}_{\text{$n$ groups of $m$ times}}
= x^{mn}
\end{equation*}
Section 6.5 Going Deeper: Radicals and Fractional Exponents
At some point in the past, you might have remembered learning that \(x^{\frac{1}{2}} = \sqrt{x}\text{.}\) Most likely, you learned this as another rule. However, you have enough knowledge at this point to learn why this is the most (and perhaps only) sensible thing that the fractional exponent could represent.
Before we take a look at fractional exponents, we will do a quick review of radicals. The first radical that students encounter is the square root. The symbol represents the non-negative number that has the property that \(\left( \sqrt{x} \right)^2 = x\text{.}\) Certain integers are considered to be perfect squares because their square roots are themselves integers. For example, we have \(\sqrt{4} = 2\) (since \(2^2 = 4\)) and \(\sqrt{25} = 5\) (since \(5^2 = 25\)). Square roots of other numbers exist, and we can either represent the exact values symbolically (such as being the number such that \(\left( \sqrt{2} \right)^2 = 2\)) or we can use a decimal approximation (\(\sqrt{2} \approx 1.41421\) because \(1.41421^2 = 1.9999899241 \approx 2\)).
An important feature of square roots is that we always take the value to be non-negative. When we think about numbers that have the property that squaring them gives you the value 4 we quickly realize that there are two possible values: 2 and -2. and This creates a potential ambiguity in our notation, because could theoretically be one of two values. However, mathematicians have adopted the convenient convention that the square root function is always non-negative.
This framework clashes with what some students have learned in their previous algebra experiences. Some students have been taught that \(\sqrt{4} = \pm 2\text{.}\) Unfortunately, this is not a proper understanding of the symbols. When we write the only value this can represent is and it never represents the value The distinction comes down to understanding what question is being asked:
- What number does \(\sqrt{4}\) represent? The value of \(\sqrt{4}\) is 2, so that \(\sqrt{4} = 2\text{.}\)
- What numbers have the property that \(x^2 = 4\text{?}\) The value can be either 2 or -2, which we can denote as \(x = \pm 2\text{.}\)
In the first case, we’re given a specific mathematical expression and are asked to determine its value. In the second case, we are given a mathematical equation and are asked to solve it. These are two different questions, which is why we end up with two different answers.
The reason that there are two solutions to \(x^2 = 4\) is because squaring a negative number results in a positive number. In fact, raising a negative number to any even power leads to a positive result. But a negative number raised to an odd power remains odd. This observation is helpful for defining higher roots.
Definition 6.5. The \(n\)-th root.
The \(n\)-th root of \(x\), denoted \(\sqrt[n]{x}\) is the number that has property \(\left( \sqrt[n]{x} \right)^n = x\text{.}\) If \(n\) is even, we pick this value to be a nonnegative number (by convention). If \(n = 2\text{,}\) we often omit the \(n\) in the notation, so that \(\sqrt{x} = \sqrt[2]{x}\text{.}\)
We can take odd roots of both positive and negative numbers, so that \(\sqrt[3]{8} = 2\) and \(\sqrt[3]{-8} = -2\text{.}\) However, we can only take even roots of nonnegative numbers and the roots are only ever nonnegative values. So \(\sqrt[4]{81} = 3\) even though both \(3^4 = 81\) and \((-3)^4 = 81\text{.}\) And it turns out that doesn’t exist because it’s impossible for a number raised to the fourth power to be negative.
This takes us back to the concept of fractional exponents. The key is to think about them in terms of properties. If we expect the properties of exponents to be consistent, then we must have the following:
\begin{equation*}
\left( x^{\frac{1}{2}} \right)^2 = x^{\frac{1}{2} \cdot 2} = x^1 = x
\end{equation*}
This means that whatever \(x^{\frac{1}{2}}\) is, it really ought to have the property that when you square it, you get \(x\text{.}\) And we have already seen that this is the definition of \(\sqrt{x}\text{,}\) which tells us that \(x^{\frac{1}{2}} = \sqrt{x}\text{.}\)
We can use the exact same logic to determine the value of \(x^{\frac{1}{3}}\text{.}\) Since \(\left( x^{\frac{1}{3}} \right)^3 = x\text{,}\) we see that \(x^{\frac{1}{3}} = \sqrt[3]{x}\text{.}\) In fact, we can see that \(x^{\frac{1}{n}} = \sqrt[n]{x}\text{.}\)
What can we say about \(x^{\frac{m}{n}}\text{?}\) Once again, we’re going to look at the properties of exponents. Notice that
\begin{equation*}
x^{\frac{m}{n}} = \left( x^{\frac{1}{n}} \right)^m = \left( \sqrt[n]{x} \right)^m
\end{equation*}
and
\begin{equation*}
x^{\frac{m}{n}} = \left( x^m \right)^\frac{1}{n} = \sqrt[n]{x^m}.
\end{equation*}
These calculations give us the ability to evaluate every fractional exponent.
Definition 6.6. Fractional Exponents.
For any integer \(m\) and positive integer \(n\text{,}\) we define \(x^{\frac{m}{n}}\) to be \(\left( \sqrt[n]{x} \right)^m = \sqrt[n]{x^m}\text{.}\)
A key observation about this definition is that we have set it up so that it would be consistent with the product rule and power rule (Theorem 6.2). It is also consistent with negative exponents (Definition 6.3). This consistency means that there are no new "rules" to learn. This is how mathematicians like to generalize results. It would be very confusing if we had one set of rules that only worked with integer exponents, and then an entirely different set of rules for working with fractional exponents. It makes much more sense to work towards consistent notation and consistent structures.
