2.4 Negative Exponents

Introduction

Building on our experience using definition of an exponent to find rules for positive powers, in this section we'll look into zero powers and negative exponents.

Then, equipped with those new tools, we'll discuss a special notation used in science that allows us to describe both values of immense measure and values of minuscule size with the same level of precision.

The Zero Exponent Rule

Let's start this section by reminding you of a very basic rule of arithmetic that we've used many times before: anything (except \(0\)) divided by itself equals \(1\).

You're used to using that rule to simplify expressions with numbers like \(\frac{13}{13}=1\), but you may not have realized it works for any type of expression, provided it doesn't equal \(0\). For instance, \(\frac{2x+3}{2x+3}=1\) as long as \(2x+3 \neq 0\) (since division by \(0\) is undefined).

So then, what can we make of an expression like \(\frac{b^3}{b^3}\)? As long as \(b \neq 0\), this has to equal \(1\).

\[\frac{b^3}{b^3}=1\]

That seems simple enough, but how does it help us find a rule for a zero exponent?

In the last section we learned about the quotient rule. Look at what happens when we simplify the same expression \(\frac{b^3}{b^3}\) using the quotient rule.

\begin{align} \frac{b^3}{b^3} &= b^{3-3} && \small \color{#5fa2ce} {\text{Use the quotient rule: } \frac{b^m}{b^n} = b^{m-n}} \\ &= b^0 && \small \color{#5fa2ce} {\text{Simplify}} \end{align}

Putting the two answers together we see that \(b^0=1\). This is our rule for a zero exponent and is true as long as \(b\) is not zero.

Just in case you were wondering, \(0^0\) does not have a defined value. It's considered "indeterminate" and just happens to be one of the things studied in a calculus course.

The Negative Power Rule

The strategy of simplifying expression two different ways and equating the results is a major way identities and rules are discovered in math. We just used that method to uncover the rule for a zero power, and we will continue to use it to find rules for negative powers.

Consider, for example, the expression \(\frac{b^4}{b^7}\). One way to simplify this is to use the definition of an exponent.

\begin{align} \frac{b^4}{b^7} &= \frac{b \cdot b \cdot b \cdot b}{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b} && \small \color{#5fa2ce} {\text{Use the definition of an exponent}} \\ &= \frac{b}{b} \cdot \frac{b}{b} \cdot \frac{b}{b} \cdot \frac{b}{b} \cdot \frac{1}{b \cdot b \cdot b} && \small \color{#5fa2ce} {\text{Use the commutative property of multiplication}}\\ &= 1 \cdot 1 \cdot 1 \cdot 1 \frac{1}{b \cdot b \cdot b} && \small \color{#5fa2ce} {\text{Since } \frac{b}{b}=1} \\ &= \frac{1}{b \cdot b \cdot b} \end{align}

Now watch what happens when we simplify the same expression using the quotient rule for exponents.

\begin{align} \frac{b^4}{b^7} &= b^{4-7} && \small \color{#5fa2ce} {\text{Use the quotient rule}} \\ &= b^{-3} && \small \color{#5fa2ce} {\text{Simplify}} \\ \end{align}

Putting the answers together, we have \(b^{-3}=\frac{1}{b^3}\). This is an example of the rule for negative powers. Simply put, \(b^{-n}=\frac{1}{b^n}\), provided \(b\neq 0\).

Whenever we are asked to "simplify" an expresssion with negative exponents, it's standard practice to rewrite the expression so that the final answer only has positive exponents.

Occasionally we might come across and expression that has a negative exponent on the bottom of a fraction, something like \(\frac{1}{b^{-3}}\). How should we deal with that? A careful application of the zero exponent rule and the quotient will give the answer.

\begin{align} \frac{1}{b^{-3}} &= \frac{b^0}{b^{-3}} && \small \color{#5fa2ce} {\text{Since } b^0=1} \\ &= b^{0-(-3)} && \small \color{#5fa2ce} {\text{Use the quotient rule}} \\ &= b^3 && \small \color{#5fa2ce} {\text{Simplify}} \\ \end{align}

How about that? The result had a positive exponent! In general, \(\frac{1}{b^{-n}}=b^n\). This is a matching counterpart to the usual rule for negative exponents. An easy way to remember both rules is to notice that a term with a negative exponent transforms into a term with a positive exponent if we shift it across the fraction line, whether it's moving from the numerator to the denominator or the other way around.

With this result in hand, we are finally ready to make a complete list of all the rules of exponents.

All of the rules we learned in the last section still apply when working with negative exponents. In fact, though we will not prove this, the rules apply to all nonzero real number exponents.

In the next examples we will need to use several of these rules.

In the first part, you might be wondering why the \(4\) didn't get moved to the bottom. The reason is that in \(4a^{-5}\) it's only the \(a\) that has a negative exponent. If it had been \((4a)^{-5}\) then the power would have applied to both and we would have moved it all to the denominator.

Scientific Notation

We've been exploring the theoretical framework around exponents, and it's now time to step into a common application: scientific notation.

Scientific notation is a compact way of writing very small numbers and very large numbers that are too long to comfortably write normally. For instance, the distance from Earth to Jupiter is \(708,823,486,000\) meters and the nearest star, Alpha Centauri, is \(40,208,000,000,000,000\) meters away. Or, on the other end of the spectrum, diameter of a hydrogen atom is about \(0.000000000106\) meters.

Imagine how easy it would be to make a mistake copying down one of those numbers, not to mention the difficulty in adding, multiplying or dividing them.

Scientific notation makes the numbers easier to write and easier to compare, by separating them into two parts: a number and a power of \(10\). The number part always has exactly one nonzero digit on the left of the decimal. This is multiplied by an appropriate power of \(10\) to make the new format equal to the original number.

It sounds more complicated than it actually is. This is one of those things that's easier to see than to read about. The following numbers are written in standard decimal format and in scientific notation.

Again, there are two key things to notice. First, the number part must be a decimal number between \(1\) and \(10\). If your starring number isn't between \(1\) and \(10\), then you move the decimal place until it is. Second, count the number of places, \(n\), that you moved the decimal point and multiply the decimal number by \(10\) raised to a power of \(n\). The sign of \(n\) should be positive if you started with a large number and negative if you started with a small number.