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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 6.1.1. 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 6.1.1. 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 6.1.2. 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.

Did you notice how Definition 6.1.1 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 6.1.3. 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.

Activity 6.1.3. 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 6.1.4. 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*}