Creatures in Flight · Quiz 31 · Rules of Exponents

The Guide

Before you begin

Quiz 29 built a power as a count of factors: 2³ means 2 × 2 × 2, three factors of 2, and the small raised number is the count. Quiz 30 ran a power backward to find a root. This quiz puts powers together and takes them apart. Powers get multiplied, divided, and raised to further powers, and each time there is a shortcut. Every shortcut in this quiz is the count of factors, kept honest. If you can count factors, you already know every rule here; the rules only save you the writing.

The shortcuts have names. When powers are multiplied, you ADD the exponents. When powers are divided, you SUBTRACT. When a power is raised to a power, you MULTIPLY. Those three words come from the Exponent Machine, a small tool that lives beside this quiz and shows each move happening; the rest stop between questions 6 and 7 opens it. The three moves come first. Then the quiz walks back down Quiz 29’s locust ladder one rung at a time, past 20¹ to 20⁰ and below, and two more names arrive on the way: ONE and FLIP.

What the raised number counts

A power has two parts. The base is the number being multiplied. The exponent, the small raised number, counts how many times the base is used as a factor. So in 2⁵ the base is 2, the exponent is 5, and the value is 2 × 2 × 2 × 2 × 2 = 32. The exponent is not a number you multiply by; it is a count of copies of the base. Keep that sentence, because every rule below is a way of counting those copies without writing them all out.

Multiplying powers: ADD

Take 2³ × 2⁴, and write both powers out. 2³ is 2 × 2 × 2, three factors. 2⁴ is 2 × 2 × 2 × 2, four factors. Multiplying them puts the two strings end to end: 2 × 2 × 2 × 2 × 2 × 2 × 2. That is seven factors of 2, which is 2⁷. Three factors and then four factors make seven factors, so the exponents were added: 3 + 4 = 7.

Three factors of 2, then four more, laid end to end, are seven factors of 2. The exponents add because they are counts, and joining two counts is adding.

Check it with the numbers themselves, which is always allowed. 2³ = 8 and 2⁴ = 16, and 8 × 16 = 128. And 2⁷ = 128. The shortcut and the long way agree, and they always will, because they are one count written two ways.

The condition: the same base. The factors could be joined into one string because they were all 2s. In 2³ × 5², the three 2s and the two 5s are different factors, and there is no single power to write them as. The Exponent Machine has a name for this case too: CAN’T COMBINE. When the bases differ there is no shortcut. Work out each power and multiply the results: 2³ × 5² = 8 × 25 = 200, and that is the end of it.

Rosa multiplies the little numbers. She writes 2³ × 2⁴ = 2¹², because a multiplication sign stands between the powers and 3 × 4 = 12. It is her habit exactly: numbers paired by where they sit rather than by what they are, the way she added straight across a pair of fractions in Quiz 10, measured a discount against the sale price in Quiz 18, and read 5:1 as one fifth in Quiz 23. Here the two small numbers sit on either side of a × sign, so she multiplies them. But they are counts of factors, and joining two counts is adding. The check that catches it: 8 × 16 = 128, and 2⁷ = 128, while 2¹² = 4,096. Her answer is thirty-two times too big.

Dividing powers: SUBTRACT

Now take 2⁵ ÷ 2² and write it as a fraction, because the fraction bar means divide: five factors of 2 on top, two factors of 2 on the bottom. A 2 over a 2 is 1, so each 2 on the bottom cancels one 2 on top. Two pairs cancel, and three factors of 2 are left on top: 2³. Five factors less two factors is three factors, so the exponents were subtracted: 5 − 2 = 3.

Each 2 on the bottom cancels one 2 on top. Two pairs go, three factors stay, and 5 − 2 = 3 is the count of what stayed.

Check with numbers: 2⁵ = 32 and 2² = 4, and 32 ÷ 4 = 8, which is 2³.

Now turn it over: 2² ÷ 2⁵. Two factors on top, five on the bottom. The same two pairs cancel, and this time three factors of 2 are left on the bottom, so the value is 1 over 2³, which is 18. Check: 4 ÷ 32 = 18. The rule, followed without looking, says 2^{2 − 5} = 2⁻³. So a negative exponent must mean what the picture shows: three factors of 2 on the bottom of a fraction. Hold onto that. The ladder below arrives at the same place from the other side.

The same canceling, with the fraction turned over. Now the three survivors are on the bottom, so the value is 1 over 2³. That is what 2⁻³ means.

A power of a power: MULTIPLY

Take (2³)². The outside exponent says: use 2³ as a factor twice. So (2³)² = 2³ × 2³, and ADD makes that 2^{3 + 3} = 2⁶. Two groups of three factors are six factors, and 3 × 2 = 6. That is the whole of MULTIPLY: it is ADD done in groups. (2³)⁴ would be four groups of three factors, twelve factors, 2¹².

The outside exponent counts groups; the inside exponent counts factors in each group. Groups times factors per group is the total, which is why this move multiplies.

Check: 2³ = 8, and 8² = 64, and 2⁶ = 64.

Three moves, told apart. Ask what is being done to the powers, not to the exponents. Powers multiplied: ADD the exponents. Powers divided: SUBTRACT. A power raised to a power: MULTIPLY. In every case the move on the exponents is one step gentler than the move on the powers, because the exponents are only counting. The Exponent Machine keeps the three on one line: multiply → ADD · divide → SUBTRACT · power of a power → MULTIPLY.

The ladder down: ONE and FLIP

Quiz 29 climbed a ladder with the desert locust. Each generation is about twenty times the one before, so after one generation a swarm is 20¹ = 20 times its old size, after two it is 20² = 400 times, and after three it is 20³ = 8,000 times. Going up a rung multiplies by 20 and adds 1 to the exponent. So going down a rung must divide by 20 and take 1 off the exponent, which is SUBTRACT with a 20¹ underneath. Start at the top and walk down. 8,000 ÷ 20 = 400, and the exponent drops from 3 to 2. 400 ÷ 20 = 20, and the exponent drops to 1. Now take one step more than Quiz 29 did. 20 ÷ 20 = 1, and the exponent drops to 0. So 20⁰ = 1. Keep going. 1 ÷ 20 is one twentieth, 120, and the exponent drops to −1, so 20⁻¹ = 120. One more: 120 ÷ 20 = 1400, and the exponent is −2, so 20⁻² = 1400, which is 1 over 20².

Every rung is got the same way, by dividing by 20 once, and the exponent keeps counting down through zero. Nothing new happens at zero except that the count passes it.

ONE, by a second road. The Exponent Machine reaches 20⁰ another way, and it is worth having both. 20³ ÷ 20³ is a number divided by itself, which is 1; you do not need to know that 20³ is 8,000 to know that. And SUBTRACT says 20³ ÷ 20³ = 20^{3 − 3} = 20⁰. One expression, two values written down, 1 and 20⁰, so they are the same number. Any base to the power 0 is 1 for the same reason: it is some power of that base divided by itself. (0⁰ is left undefined, and the test does not ask about it.)

FLIP: what a negative exponent is. Two roads have now reached one place. The ladder gave 20⁻² = 1400, which is 1 over 20², and the turned-over fraction gave 2⁻³ as 1 over 2³. A negative exponent is a count of factors on the bottom of a fraction. The Exponent Machine’s name for the move is FLIP: a power with a negative exponent moves across the fraction bar, and its exponent turns positive. So 5⁻² = 1 over 5² = 125, and 1 over 3⁻¹ is 3¹ = 3. The sign belongs to the exponent, and it says which side of the bar the factors are on. It never makes the value negative: 2⁻³ is one eighth, a small positive number, and not negative eight.

Andre reads the sign as the answer’s. He writes 2⁻³ = −8, because he sees a minus sign, a 2 and a 3, and makes a negative number out of them. It is his habit: he reads size off the digits without asking what place they are in, the way a bigger denominator looked bigger to him in Quiz 9 and again in decimals in Quiz 13, the way 0.5% looked like a half in Quiz 17, and the way −8 looked bigger than −3 in Quiz 20. Here the minus sign is in the exponent’s place, and in that place it means on the bottom, not below zero. The check: by ADD, 2³ × 2⁻³ = 2⁰ = 1. With the right value, 8 × 18 = 1. With Andre’s, 8 × (−8) = −64. Only one of those is 1.

Where the starlings come in

The photograph at the top of this page is a flock of starlings over the marshes of southern Denmark. When a flock like this turns, it turns almost all at once. A team in Rome filmed flocks of a few thousand starlings with several cameras at the same moment, worked out where every bird was, and found that each bird keeps track of about 7 of its nearest neighbors, however near or far those 7 happen to be. That finding is a measurement, published in 2008, and the gallery at the foot of this page says more about it.

What follows is a simplified model, not a measurement, and it matters which is which. Start with 1 bird that turns, and suppose that each bird that turns is followed, one step later, by 7 others. Then 7 birds turn at the first step, 7 × 7 = 49 turn at the second, and 7 × 7 × 7 = 343 turn at the third. Written as powers, that is 7¹, 7², 7³: the exponent is the number of steps the turn has traveled. Real flocks do not work this neatly. Neighbors overlap, a bird cannot turn twice, the flock runs out of birds, and the turn travels through it as a wave. But the model gives the three moves a place to stand.

The bird and its 7 neighbors are the measured part. The row of counts, 1, 7, 49, 343, is the model: how many birds turn at each step. The exponent is the number of steps the turn has traveled.

The move	In the model	The count
ADD	At its 2nd step a turn has 7² birds turning. Each of those passes it on for 3 more steps, and 3 steps is 7³ birds for each one.	7² × 7³ = 7⁵ = 16,807 birds turning at the 5th step
SUBTRACT	16,807 birds are turning at one step, and 49 were turning at an earlier one. How many steps apart?	7⁵ ÷ 7² = 7³, so 3 steps
MULTIPLY	A ripple of 2 steps, and each bird at its end starts another 2-step ripple, and again: 3 ripples of 2 steps.	(7²)³ = 7⁶ = 117,649 birds turning at the 6th step

The chart, and the two names still to come

Open the chart: the three moves, and the names they earn

Name	When	The move	Check with numbers
ADD	multiplying powers, same base	2³ × 2⁴ = 2⁷	8 × 16 = 128 = 2⁷
SUBTRACT	dividing powers, same base	2⁵ ÷ 2² = 2³	32 ÷ 4 = 8 = 2³
MULTIPLY	a power of a power	(2³)² = 2⁶	8 × 8 = 64 = 2⁶
ONE	the zero exponent	20⁰ = 1	20 ÷ 20 = 1
FLIP	negative exponents	2⁻³ = 1 over 2³	1 ÷ 8 = 0.125
CAN’T COMBINE	different bases	2³ × 5² stays as it is	8 × 25 = 200

The chart is wider than your screen: slide it sideways with your finger, or turn your phone.

A one-page printable version, the Rules of Exponents chart, is in the same folder as this quiz: exponent-rules-chart.html. It also carries two names this quiz has not used, DISTRIBUTE and ROOT, which belong to Quiz 31a.

Worked Examples

Three to study before you start

Example 1 · Multiplying powers: count, then ADD, then check

Find 3² × 3³. Count the factors first: 3² is two 3s and 3³ is three 3s, so together there are five 3s, which is 3⁵. Now the shortcut: same base, powers multiplied, so ADD the exponents: 3^{2 + 3} = 3⁵. Its value is 3 × 3 × 3 × 3 × 3 = 243. Check with the numbers: 3² = 9 and 3³ = 27, and 9 × 27 = 243. The count, the rule and the arithmetic all say the same thing, which is what you should expect, because they are one fact three ways.

Example 2 · Dividing powers, right side up and upside down

Find 5⁴ ÷ 5¹. As a fraction: four 5s on top, one 5 on the bottom. The one 5 below cancels one above, and three 5s are left on top: 5³. By SUBTRACT: 5^{4 − 1} = 5³ = 125. Check: 5⁴ = 625, and 625 ÷ 5 = 125.

Now upside down: 5¹ ÷ 5⁴. One 5 on top, four on the bottom. The pair cancels, and three 5s are left on the bottom, so the value is 1 over 5³, which is 1125. By SUBTRACT: 5^{1 − 4} = 5⁻³, and the fraction picture has just told you what that means: three factors on the bottom, 1125. Check: 5 ÷ 625 = 0.008, and 1 ÷ 125 = 0.008. Same number, one written as a fraction and one as a decimal.

Example 3 · A power of a power, then two moves in one problem

Find (5²)³. Read the outside exponent as a count of groups: three groups of 5², which is 5² × 5² × 5². Three groups of two factors is six factors. By MULTIPLY: 5^{2 × 3} = 5⁶ = 15,625. Check: 5² = 25, and 25 × 25 = 625, and 625 × 25 = 15,625.

Now (3²)² × 3. Two moves, and the order is the one Quiz 6 gave: what is inside parentheses first, then the power of a power, then the multiplication. MULTIPLY first: (3²)² = 3⁴. Then ADD: the lone 3 is 3¹, so 3⁴ × 3¹ = 3⁵ = 243. Check: 3⁴ = 81, and 81 × 3 = 243. Notice the 3 with no exponent showing. Every plain number is a power with exponent 1, and writing that 1 in is what makes the count come out right.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. Whenever a rule gives you an answer, check it once with the numbers themselves, the way the worked examples do.

Write 2³ × 2² as a single power, then find its value.
Write 5⁶ ÷ 5⁴ as a single power, then find its value.
Rosa worked out 2³ × 2⁴ and wrote 2¹². Which statement names what she did?
- A) She worked out 2³ = 8 and then added the 4, getting 12.
- B) She should have multiplied the bases as well, getting 4¹².
- C) She multiplied the exponents because a multiplication sign stood between the powers. The powers share a base, so the exponents are added: 2⁷.
- D) Nothing: 2¹² is correct, because multiplying powers means multiplying their exponents.
Write (3²)³ as a single power, then find its value.
Here are three expressions: 4³ × 4², then 4³ × 5², then 4³ ÷ 4³. For each one, say whether its exponents can be combined into one power, and give its value.
On the locust ladder, 20² = 400 and 20¹ = 20. Walk two more rungs down. Give 20⁰ and 20⁻¹, and say how each one is got from the rung above it.

ToolA rest stop with the machine running, for the stubborn ones.

Open the Exponent Machine, which sits in the same folder as this quiz, at thepeoplesshare.org/ged/math/exponent-machine.html. Tap a tile to choose a move, set your own base and exponents, and watch the move happen, with the voice on or off. Two places to land: SUBTRACT, where dividing 2³ by 2⁵ leaves two 2s stranded on the bottom of the fraction, and ONE, which reaches the zero exponent by both roads at once.

The picture above is drawn for this page in the Machine’s colors; it is not a screen from the Machine itself.

Write 10⁻³ as a fraction and as a decimal.
In the simplified flock model, a turn that has traveled 2 steps has 7² birds turning. Each of those birds passes the turn on for 3 more steps, and 3 steps is 7³ birds for each one. Which single power gives the number of birds turning at the 5th step?
- A) 7⁵, because 7² × 7³ adds the exponents.
- B) 7⁶, because 7² × 7³ multiplies the exponents.
- C) 14⁵, because the bases are added along with the exponents.
- D) 7² + 7³, because the second ripple is added to the first.
Find the value of (2⁴ × 2³) ÷ 2⁵.
Andre says that 2⁻³ = −8. Say what the minus sign in the exponent asks for, and give the value of 2⁻³ as a fraction.

Answer Key

Check your work

Open the key: after you've finished all ten

1 2⁵ = 32

Same base, powers multiplied, so ADD: 3 + 2 = 5 factors of 2, and 2⁵ = 32. Check with the numbers: 2³ = 8, 2² = 4, and 8 × 4 = 32. If you wrote 2⁶ = 64, you multiplied the exponents; if you wrote 4⁵ = 1,024, you multiplied the bases as well. Both are answered by counting: three 2s and two more 2s are five 2s.

2 5² = 25

Same base, powers divided, so SUBTRACT: 6 − 4 = 2 factors of 5 left on top, and 5² = 25. Check: 5⁶ = 15,625 and 5⁴ = 625, and 15,625 ÷ 625 = 25. The check is longer than the rule, which is the point of the rule; but it is the check that tells you the rule was used rightly.

3 C.

2³ × 2⁴ is three 2s and then four 2s, seven in all, so 2⁷ = 128, and 8 × 16 = 128 agrees. Rosa’s 2¹² is 3 × 4 in the exponent, the two small numbers paired because a × sign sat between them, which is her habit from Quizzes 10, 18 and 23. A) describes arithmetic nobody did, though 8 + 4 happens to give 12, which is what makes it tempting. B) would make the error worse: 4¹² is 16,777,216. D) states her error as though it were the rule, which is how an error survives.

4 3⁶ = 729

A power of a power, so MULTIPLY: three groups of two 3s is six 3s, and 2 × 3 = 6. Then 3⁶ = 729. Check: 3² = 9, and 9 × 9 × 9 = 729. If you wrote 3⁵ = 243, you added where the outside exponent was counting groups.

5 4⁵ = 1,024; cannot be combined, and equals 1,600; 4⁰ = 1

4³ × 4² shares a base, so ADD: 4⁵ = 1,024 (check: 64 × 16 = 1,024). 4³ × 5² does not share a base, so there is no single power to write; work each out and multiply: 64 × 25 = 1,600. If you wrote 20⁵ or 9⁵, you combined what cannot be combined, and 20⁵ is 3,200,000, two thousand times too big. 4³ ÷ 4³ is a number divided by itself, which is 1, and SUBTRACT gives 4⁰, so 4⁰ = 1: both roads to ONE in one line.

6 20⁰ = 1 and 20⁻¹ = 120

Each rung down divides by 20 and takes 1 off the exponent. From 20¹ = 20: 20 ÷ 20 = 1, and the exponent drops from 1 to 0, so 20⁰ = 1. From there: 1 ÷ 20 = 120, and the exponent drops to −1, so 20⁻¹ = 120, which is 0.05 as a decimal. If you wrote 20⁰ = 0, the ladder answers you: the rung above is 20, and 20 ÷ 20 is not 0.

7 11,000 = 0.001

A negative exponent counts factors on the bottom: 10⁻³ is 1 over 10³, which is 1 over 1,000. As a decimal that is 0.001, one thousandth, three places to the right of the point, which is Quiz 13’s place value and Quiz 15’s powers of ten meeting FLIP. If you wrote −1,000, the sign was read as belonging to the answer; it belongs to the exponent, and it says on the bottom.

8 A.

7² birds, each starting 7³ more, is 7² × 7³, and powers multiplied ADD their exponents: 7⁵ = 16,807, two steps and then three more is five steps. B) multiplies the exponents, Rosa’s error from question 3, and 7⁶ = 117,649 is what the model gives one step later. C) adds the bases, and 14⁵ = 537,824, which is nowhere near. D) adds the two ripples instead of multiplying, and 49 + 343 = 392 would mean the second ripple added a few hundred birds instead of multiplying the first by 343. Remember also what the model is: a simplified way of counting, not a count of real birds.

9 4

Inside the parentheses first, by Quiz 6’s order: 2⁴ × 2³ = 2⁷ by ADD. Then 2⁷ ÷ 2⁵ = 2² by SUBTRACT, and 2² = 4. Check: 16 × 8 = 128, and 128 ÷ 32 = 4. Two moves, and each one is a count: seven 2s, then five of them canceled by the five below, leaving two.

10 18

The minus sign is in the exponent’s place, and there it asks for factors on the bottom of a fraction: 2⁻³ is 1 over 2³, which is 18, or 0.125. It is a small positive number. Andre’s −8 is his habit from Quizzes 9, 13, 17 and 20: size read off the digits without asking what place the sign is in. The check from the Guide: 2³ × 2⁻³ must be 2⁰ = 1 by ADD, and 8 × 18 = 1, while 8 × (−8) = −64.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2, 4	The three moves, one at a time	Write the factors out once and count them. ADD when powers are multiplied, SUBTRACT when they are divided, MULTIPLY for a power of a power. The count is the rule.
3	Naming the error	The small numbers are counts, and joining two counts is adding. Check any rule with the numbers themselves; a wrong rule fails the check every time.
5	Same base, or not	The shortcuts join factors that are alike. With different bases, work each power out and multiply. A power divided by itself is 1.
6, 7, 10	The ladder: zero and negative exponents	Each rung down divides by the base and takes 1 off the exponent. The zero exponent is 1. A negative exponent puts factors on the bottom of a fraction; it never makes the value negative.
8, 9	More than one move in a problem	Parentheses first, then one move at a time, naming each as you go. Check the finished answer with the numbers.

Eight or more right: the three moves are yours, and Quiz 31a can hand you the last two names, DISTRIBUTE and ROOT. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we’ve found the right ground to work. One habit for a week: before you use any rule, write the factors out once, count them, and watch the rule come true. Then come back to these same ten.

Where this goes

Next in Creatures in Flight: Quiz 31a — Fractional Exponents. An exponent can be a fraction, and 4^1/2 turns out to be a square root. ROOT arrives, and DISTRIBUTE with it, and the quiz shows why 4^3/2 is easier to work by taking the root first, and that both orders agree. The pictures stay with creatures in flight.

The Company · An Interlude

The black sun

A great flock of starlings, thousands of small dark shapes, spread across a sunset sky above flat marshland with farm buildings on the horizon — Starlings over Tøndermarsken, in the marshes of southwest Jutland, Denmark, at sunset. Photograph by Tommy Hansen, from PDFnet.dk, which places no copyright on its pictures, by way of Wikimedia Commons.

Every spring and every fall, starlings on their way between the north and their winter grounds stop in the marshes near the Wadden Sea in the south of Denmark. Toward evening they gather in the air over the reed beds before dropping into them for the night, and the flocks can run to hundreds of thousands of birds. The Danes call the sight sort sol, the black sun, because a flock this size can put out the light. One of those evenings is the picture above, in its own colors; a copy recolored into the house colors is this quiz’s masthead.

A vast flock of starlings filling an orange evening sky above a line of bare trees, with a dark band curling through the flock where the birds are packed closest — A murmuration of starlings at Gretna, on the Scottish side of the border with England, on the evening of November 7, 2011. Photograph by Walter Baxter, from the Geograph project, shown as taken. Walter Baxter / A murmuration of starlings at Gretna / CC BY-SA 2.0.

The same thing happens wherever starlings gather in numbers to roost, and this flock is over the fields at Gretna, in the south of Scotland, on a November evening. It is folding over itself, and the dark band curling through it is where the birds are packed closest. A flock like this holds its shape while every bird in it is moving at full flying speed, and no bird can see the whole flock.

A patch of the Danish flock enlarged, so that each starling shows as a single dark mark against the orange sky — A patch of the Danish flock, enlarged. Each bird is one mark, and each keeps track of about seven of the marks nearest to it.

How a flock this size turns together was a guess until it was measured. In 2008 a team of physicists working in Rome, M. Ballerini and eleven others, published what they had found by filming starling flocks of a few thousand birds over the city from several cameras at once and reconstructing the position of every bird. Each bird, they found, keeps track of about six or seven of its nearest neighbors, however far away those neighbors are. A bird does not watch everything within a fixed distance; it watches a fixed number of birds. That is why the flock stays a flock when it stretches thin or bunches up: the seven are always there to watch. The paper is free to read, in the Proceedings of the National Academy of Sciences, and its first page says all of this in one paragraph.

The ripple in the Guide, 7, then 49, then 343, is not in that paper. It is a simplified model, built here so that the rules of this quiz would have something to count, and the way to see how simplified it is to keep counting. At the fourth step the model has 7⁴ = 2,401 birds turning, at the fifth 7⁵ = 16,807, at the sixth 7⁶ = 117,649, and at the seventh 7⁷ = 823,543, which is more birds than most flocks hold. The real flock has run out of birds long before the model has, because in a real flock the seven neighbors of one bird are also the neighbors of the birds beside it, and a bird cannot turn twice. That is what a simplified model is for. It is not the flock. It is a way of counting that lets the flock be thought about, and it tells you plainly where it stops being true.

The rules of this quiz are the same count. 7² × 7³ is not a new fact about starlings; it is 49 × 343 with the multiplying put off, and the exponent 5 is how many steps the turn has traveled. Every rule on the chart is a way of keeping that count without writing out the factors, and every one of them can be checked by writing the factors out once.

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