1.5 Working with Exponents

Learning Objectives

By the end of this section, you will be able to:

Section 0.1 introduced exponents as shorthand. Writing \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) over and over is tedious, so we write \(2^5\) instead and agree that the small raised number counts how many factors there are. That is all an exponent has ever meant in this book, and it is all it will mean in this section. Nothing new is being defined.

What is new is what happens when exponential expressions start meeting each other. What is \(2^5 \cdot 2^3\)? What is \(\dfrac{x^7}{x^4}\)? What is \((x^2)^3\)? Each of these has a short answer, and each short answer is a rule with a name. There are seven or eight of them, they look confusingly alike, and every year students memorize the list, mix up which one adds and which one multiplies, and lose points to the confusion.

So this section is going to do something different. Every rule below will be derived — expanded into repeated multiplication, counted, and collapsed back down. That takes a few extra lines, but it buys something worth much more than the lines cost: if you can rebuild a rule, you never have to remember it. A student who has forgotten whether exponents add or multiply in \((x^2)^3\) can write out \(x^2 \cdot x^2 \cdot x^2\), count six \(x\)'s, and be sure in ten seconds. A student who only memorized has to guess.

1.5.1 What an Exponent Records

Start with the vocabulary, because the rest of the section depends on saying it precisely.

The base is exactly what the exponent is touching

Here is the first place people lose points, and it is a notation issue rather than a mathematics issue.

The two expressions are not the same. In \((-5)^4\) the parentheses wrap the negative sign into the base, so four factors of \(-5\) are multiplied, and four negatives pair off into positives. In \(-5^4\) there are no parentheses, so the base is just \(5\); the exponent applies to the 5 alone, and the minus sign stays out front, waiting. Read it as "the opposite of \(5^4\)."

Order of operations settles this: exponents are evaluated before the negation, unless parentheses say otherwise.

The pattern in that table is worth stating on its own: parentheses decide the base. Whatever sits immediately inside the parentheses is what gets multiplied by itself. This will come up again with negative exponents in §1.5.5, where the same question — what exactly is the base? — separates \(5y^{-1}\) from \((5y)^{-1}\).

The base can be a fraction or a decimal, and nothing changes.

1.5.2 The Product Property: Multiplying Powers of the Same Base

Now to the first of the rules. Rather than announce it, let us look at a case and see what it forces.

What could \(x^2 \cdot x^3\) possibly be? Expand both factors and stop thinking about exponents entirely.

Two factors, then three more factors, all multiplied together — five factors total. Nothing was decided; the count decided it. And \(5\) is \(2 + 3\), which is not a coincidence: the exponents were the counts, and the counts got pooled.

The mistake to head off now

This is the single most common error in the whole section, so meet it deliberately rather than by accident. There are two halves to it, and they can appear together.

Consider \(3x^4 \cdot 5x^2\). The commutative and associative properties from §0.5 let the factors be rearranged and regrouped:

Those are two different operations happening in the same line, and the table below shows what it looks like when they get crossed. Read the third column as work you might actually see on a quiz, and the fourth as the diagnosis.

1.5.3 The Quotient Property: Dividing Powers of the Same Base

Division should undo multiplication, so if multiplying adds exponents, dividing ought to subtract them. Rather than assume that, derive it the same way — expand and count.

Consider \(\dfrac{x^5}{x^2}\).

There are two \(x\)'s downstairs and five upstairs. Each downstairs \(x\) pairs with an upstairs \(x\), and each such pair is \(\dfrac{x}{x} = 1\) — that is the disguised-1 move from §0.5.2, run backwards. Every \(x\) downstairs takes an \(x\) upstairs off the floor. Two pairs cancel to 1, and three \(x\)'s are left upstairs with nothing to pair with.

And \(3 = 5 - 2\). The subtraction is just bookkeeping on how many factors were left over.

Now run a case where the bigger pile is downstairs. Consider \(\dfrac{x^2}{x^5}\).

Two pairs cancel again, and this time three \(x\)'s are left downstairs. When every factor upstairs has been paired off, a 1 is left in the numerator — dividing a quantity out entirely leaves 1, not nothing.

Having two forms of one property is awkward, and you may already suspect it is unnecessary. It is. By the end of §1.5.5 the second form will be absorbed into the first, and the rule will read simply \(\dfrac{a^m}{a^n} = a^{m-n}\), with no case split at all. Getting there is what the next two subsections are for.

1.5.4 The Zero Exponent

The quotient property was stated for \(m > n\) and for \(n > m\). It said nothing about \(m = n\), and that gap is where something genuinely new appears.

What is \(\dfrac{a^m}{a^m}\)? There are two honest ways to answer, and they must agree.

Route one — it is a number over itself. Any nonzero quantity divided by itself is 1. So

Route two — apply the quotient property. Subtract the exponents, exactly as the property says:

Both routes describe the same expression, so both answers describe the same number.

This is worth sitting with, because \(a^0\) has no meaning under the original definition — "use \(a\) as a factor zero times" is not a picture of anything. The value 1 was not chosen because it felt nice. It is the only value that keeps the quotient property working, and that is a much stronger reason than a definition handed down.

The same conclusion arrives from a second direction, and this direction is the one to carry into §1.5.5. Write out the powers of 3 in descending order and watch what happens to the values.

Every time the exponent drops by 1, the value is divided by 3 — which makes sense, since each step removes one factor of 3. The pattern is perfectly regular for the rows we already understand, and the next step down is \(3 \div 3 = 1\). The value of \(3^0\) is not being invented; it is being forced by refusing to break a pattern that holds everywhere else.

Notice what the condition \(a \neq 0\) is doing. The expression \(0^0\) is excluded, and the table shows why: to reach it you would divide by the base at each step, and dividing by 0 is undefined. The rule holds for every base except the one base that breaks division.

Figure 1.5.4 — The zero exponent: continuing the divide-by-the-base pattern forces the value 1.

1.5.5 Negative Exponents

The descending table has no reason to stop at \(3^0\). Nothing about the step "divide by 3" runs out. So continue it and see what the pattern demands.

Reading the bottom three rows: \(3^{-1} = \dfrac{1}{3} = \dfrac{1}{3^1}\), and \(3^{-2} = \dfrac{1}{9} = \dfrac{1}{3^2}\), and \(3^{-3} = \dfrac{1}{27} = \dfrac{1}{3^3}\). A negative exponent produces the reciprocal of the matching positive power. Again, nothing was chosen — the pattern had one continuation and this is it.

The quotient property gives the same answer, which is the confirmation that matters. Take \(\dfrac{x^2}{x^5}\) and refuse to case-split. Just subtract:

But §1.5.3 already worked that same quotient out by cancelling and got \(\dfrac{1}{x^3}\). One expression, two correct computations, so the results are equal:

A negative exponent in the denominator

What if the negative exponent starts out downstairs? Apply the definition and simplify the complex fraction that results.

Identifying the base, again

The warning from §1.5.1 returns with more consequences, because now the exponent moves things around.

In the first, the exponent touches only \(y\), so only the \(y\) moves down and the 5 stays put. In the second, the parentheses make \(5y\) the base, so the whole thing moves. Same symbols, different grouping, different answers.

The full sign-and-grouping table, which repays slow reading:

A fraction raised to a negative power

The last row of that table used a shortcut that deserves its own derivation. What is \(\left(\dfrac{3}{4}\right)^{-2}\)?

Now notice that \(\dfrac{16}{9}\) is exactly \(\left(\dfrac{4}{3}\right)^2\). So the whole four-step trip amounted to one move: flip the fraction, and make the exponent positive. Every shortcut here is a derivation you already did. Flipping the fraction looks like a trick until you have walked those four steps once.

What negative exponents bought

Two things, and both are permanent.

First, the quotient property collapses to one form. There is no longer any need to check which exponent is larger before dividing — just subtract, and if the result is negative, apply the definition at the end.

Second, there is now a convention about what a finished answer looks like. An expression is not considered fully simplified while it still contains a negative exponent. If simplifying lands you on \(x^{-3}\), take the extra step and write \(\dfrac{1}{x^3}\). That final rewrite is expected on every answer in this section.

1.5.6 The Power Properties

Three more rules, all derived the same way: expand, count, collapse.

Raising a power to a power

What is \((x^2)^3\)? The outer exponent says "use the quantity \(x^2\) as a factor three times." So write it out.

Three groups of two factors each is \(3 \times 2 = 6\) factors. Here the exponents multiply, and the counting shows exactly why: this is not pooling piles of different sizes, it is copies of one pile. Same picture every time: identical groups multiply, mixed piles add.

Raising a product to a power

What is \((2x)^3\)?

Every factor inside got used three times, so every factor picks up the exponent.

Note also the limit of this property. It applies to a product inside the parentheses, not a sum. \((a + b)^2\) is not \(a^2 + b^2\) — check it with \(a = 3\) and \(b = 4\): the left side is \(7^2 = 49\) and the right side is \(9 + 16 = 25\). Chapter 7 takes up what \((a+b)^2\) actually equals.

Figure 1.5.8 — Raising a product to a power: the exponent reaches every factor inside, coefficient included.

Raising a quotient to a power

Same derivation, with division in place of multiplication.

To raise a fraction to a power, raise the numerator and the denominator to that power: \(\left(\dfrac{a}{b}\right)^m = \dfrac{a^m}{b^m}\), for \(b \neq 0\).

1.5.7 Putting the Properties Together

Everything derived so far, collected. Assume \(a\) and \(b\) are real numbers and \(m\) and \(n\) are integers, with denominators nonzero wherever one appears.

Most real problems use several of these at once. There is no single required order, but a reliable habit is: clear the parentheses first (power properties), then combine like bases (product and quotient), then clean up negatives and zeros last.

Now with negative exponents in play. Nothing about the order changes — parentheses first, like bases next, negatives last — but the last stage now has real work in it.

And one that needs nearly everything at once.

A final collection of the errors this section is designed to prevent:

1.5.8 Scientific Notation

Everything above was algebra. This last part is what the algebra is for when the quantities get out of hand.

Some numbers are unwieldy to write. A number in the millions carries a long tail of zeros; a number in the millionths carries a long head of them. Miscounting those zeros is easy, comparing two such numbers by eye is unreliable, and multiplying them by hand is worse. Because our number system is built on tens, powers of ten give a compact way out.

Take 4,000 and 0.004 and rewrite each using place value.

The second row is where the negative-exponent work pays off directly. Without §1.5.5, \(10^{-3}\) would be a symbol with no meaning; with it, \(10^{-3}\) is exactly \(\dfrac{1}{1000}\), and small numbers get written as compactly as large ones.

Decimal form to scientific notation

1. Move the decimal point so that exactly one nonzero digit sits to its left — that gives the factor \(a\).
2. Count how many places \(n\) the point moved.
3. If the original number was greater than 1, write \(\times 10^{n}\). If it was between 0 and 1 the quantity is small, so the power of ten will be negative: write \(\times 10^{-n}\).
4. Check by carrying out the multiplication.

More conversions, worked the same way:

The exponent is not arbitrary — it records the size of the number. A positive exponent means "large," a negative exponent means "small," and the size of the exponent says how large or how small.

Scientific notation to decimal form

1. Read the exponent \(n\) on the 10.
2. Move the decimal point \(n\) places — right if \(n\) is positive, left if \(n\) is negative.
3. Fill in zeros as placeholders wherever the point moved past the end of the digits.
4. Check.

Comparing sizes

One practical payoff: scientific notation makes it fast to sort quantities that differ wildly in size, because the exponent does most of the comparing. Compare exponents first; only when two numbers share an exponent do you need to look at the leading factors.

A biologist collects bird population estimates from eight parks, some already in scientific notation and some not. Putting them all in the same form makes the ordering readable.

The rows are already in order from least to greatest. Reading the exponents did almost all of that sorting: everything with \(10^2\) comes before everything with \(10^3\), and everything with \(10^3\) comes before everything with \(10^4\). Three exponents were shared, and there the leading factors had to settle the order: \(10^{3}\) by Daniels and Shady Elm, \(10^{4}\) by Blue Spruce, Red Elk and Yellow Jay, and \(10^{5}\) by Westcott and Bear. Every one of those comparisons is between numbers under 10 — easier than comparing the full populations.

Multiplying and dividing in scientific notation

The exponent properties apply to powers of ten exactly as they apply to powers of \(x\). Nothing about a base of 10 is special. That is what makes arithmetic on very large and very small numbers manageable.

To multiply, group the leading factors together and the powers of ten together — the commutative and associative properties from §0.5.1 — then use the product property on the tens.

Sometimes the leading factors multiply to something 10 or larger, which breaks the \(1 \leq |a| < 10\) requirement. One extra step fixes it: rewrite the oversized factor in scientific notation itself and combine the powers of ten again.

To divide, do the same thing with the quotient property.

In every case the leading factors are handled by ordinary arithmetic and the powers of ten by the exponent properties. The two halves never interfere with each other, which is exactly the separation §1.5.2 insisted on when it said that coefficients multiply while exponents add.

1.5.9 Why This Section Comes Before Chapter 4

Chapter 4 studies exponential functions: relationships in which a quantity is repeatedly multiplied by the same factor, written \(y = a \cdot b^{t}\). The starting amount is \(a\), the growth factor is \(b\), and the variable \(t\) sits in the exponent. Three examples, all of which Chapter 4 will treat properly.

Growth. A marine biologist measures a coral structure at 1,200 cubic centimeters and finds that its volume doubles each year. The relationship is

where \(t\) is years since the measurement and \(y\) is volume in cubic centimeters. Substituting \(t = 5\) gives \(1200 \cdot 32 = 38{,}400\) cubic centimeters five years later. Substituting \(t = 0\) gives \(1200 \cdot 2^0 = 1200 \cdot 1 = 1200\), which is the original measurement. The zero exponent property is what makes \(t = 0\) return the starting value.

Reading backwards in time. Negative values of \(t\) are where §1.5.5 stops being abstract. In this model \(t = -1\) means one year before the biologist measured, and

which says the coral was 600 cubic centimeters a year earlier — sensible, since it doubles each year. Likewise \(t = -2\) gives \(1200 \cdot \dfrac{1}{4} = 300\). The negative exponent is not a strange edge case; it is how the model talks about the past.

That closes more than this section. §1.1 generalized one border into \(4N + 8\), an expression that works for a center square of any size. §1.2 turned a growing pattern into a table and then into two rules, \(N(t) = 2 \cdot N(t-1)\) and \(N(t) = 2^{t}\). §1.3 gave that relationship a name — function — and a notation, \(f(x)\), for asking it questions without redrawing the picture. §1.4 asked what a function's inputs and outputs are allowed to be, and answered with domain and range, discrete or continuous. §1.5 supplied the algebra underneath all of it — the rules that let \(2^{t}\), \(8 \cdot 2^{t}\) and \(60(0.8)^{t}\) be trusted instead of merely used.

Chapter 2 opens on linear functions — a new kind of rule to name, evaluate, and bound the domain of, using exactly the vocabulary this chapter built. When Chapter 4 later asks what happens to a population every year forever, the question will be about growth, not about whether exponents add or multiply. That part is settled now, and settled in a way you can rebuild from scratch any time you need to.

Problem Set 1.5

Key Terms