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An 1859 engraving of a locust swarm: winged locusts fill the air and the foreground while a man swings a flail at them in front of a farmhouse An 1859 engraving of a locust swarm: winged locusts fill the air and the foreground while a man swings a flail at them in front of a farmhouse
The People's Share
Creatures in Flight · Quiz 29

Exponents

The small raised number, what it counts, and how fast it runs
A colored engraving of a locust swarm by Emil Schmidt, printed in the German weekly Die Gartenlaube in 1859; the man at the left is beating at the swarm with a flail. Public domain, from Wikimedia Commons. Printed here in the house colors; the gallery at the foot of the page shows it as it was drawn.
The Guide

Before you begin

An exponent is a short way of writing the same number multiplied by itself several times. That is the whole idea, and it is a small one. The trouble is that the writing is small too: the number that does the counting sits raised at the shoulder, in smaller type, and it is easy to read it as something to multiply by. This quiz is about reading that raised number correctly, computing what it stands for, knowing the squares by sight, and keeping exponents in their place when other operations are in the room. The GED asks for all of that, and it asks it in the middle of longer problems, which is why the reading has to be automatic.

What the raised number counts

Write 53 and you have said: multiply three fives together. It is 5 × 5 × 5 = 125. The 5 is called the base, and the raised 3 is called the exponent. The exponent counts how many times the base appears in the multiplication. It is not something the base gets multiplied by. Five times three is 15, and 15 has nothing to do with 53.

The guard for a power: write the factors out before you compute. For 63, write 6 × 6 × 6. If the written line does not have as many copies of the base as the exponent says, the exponent was misread. This costs a few seconds and catches the one error that matters most here.

Squared and cubed, and where those words come from

An exponent of 2 is read squared and an exponent of 3 is read cubed, and the words are literal. A square measuring 3 on each side holds 3 × 3 = 9 unit squares, which is Quiz 27's area. A cube measuring 3 on each edge holds 3 × 3 × 3 = 27 unit cubes. So 32 is the area of that square and 33 is the volume of that cube, and the shapes are where the names came from.

That difference between 9 and 27 is worth sitting with, because it is the reason a living thing cannot simply be made bigger. Double every length of an animal and its skin and its bones' cross sections grow by 22, four times over, while its weight grows by 23, eight times over. Quiz 24's gallery said an ant cannot simply be scaled up, and Quiz 25 and Quiz 27 said it again with areas. This is the arithmetic underneath it: two exponents on the same measurement, pulling apart. Quiz 30 runs that arithmetic backward.

The perfect squares, by sight

A perfect square is what you get when a whole number is squared: 1, 4, 9, 16, 25 and on up. They are the gold diagonal of Quiz 4's multiplication table, where each row meets its own column. They are worth knowing by sight through 144, because the test uses them constantly and because Quiz 30 undoes them.

Open the multiplication table: the perfect squares are the gold diagonal
×123456789101112
1123456789101112
224681012141618202224
3369121518212427303336
44812162024283236404448
551015202530354045505560
661218243036424854606672
771421283542495663707784
881624324048566472808896
9918273645546372819099108
10102030405060708090100110120
11112233445566778899110121132
121224364860728496108120132144

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

The gold cells, read in order: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Each one is a whole number times itself.

Powers of ten

Ten is the friendliest base there is, because its powers are written into the way numbers are spelled. 102 = 100, 103 = 1,000, 106 = 1,000,000. The exponent counts the zeros after the 1, and it counts them because each factor of ten moves every digit one place to the left, which is the place-value slide from Quiz 15. Powers of ten are how very large numbers get written down without a row of zeros nobody can count, and that is a topic of its own later in this leg.

Exponents keep their place in the order of operations

Quiz 6 set the order: whatever is inside parentheses, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Exponents come second, before multiplying, which is the step most often lost.

So 3 + 4 × 22 is worked in this order: 22 = 4, then 4 × 4 = 16, then 3 + 16 = 19. Adding first would give (3 + 4) × 4 = 28, and squaring the whole left side would give something else again. The order is not a preference; it is what the writing already means.

Where the minus sign lives. Since Quiz 22, a negative times a negative is positive, and exponents inherit that. But the parentheses decide what the base is. In (−5)2 the parentheses put the minus inside the base, so it is (−5) × (−5) = 25. In −52 the exponent belongs to the 5 alone and the minus sign stays outside, so it is the opposite of 5 × 5, which is −25. Two answers, one sign of difference, and the parentheses are the whole of it.

A number that outruns counting

The desert locust breeds fast enough that its numbers are usually reported as a multiplication rather than a count. The Food and Agriculture Organization of the United Nations, which watches locusts for a living, publishes the increase by generation: about 20 times as many after 3 months, 400 times after 6 months, and 8,000 times after 9 months.

Those three figures are one exponent counted up. A generation takes about 3 months and is about 20 times the one before, so after 1 generation the swarm is 201 = 20 times its old size, after 2 generations it is 202 = 400 times, and after 3 generations it is 203 = 8,000 times. The exponent is not counting locusts. It is counting generations, and the base is what one generation does.

Worked Examples

Three to study before you start

Example 1 · Reading a power, and computing it

Find 43. Write the factors out first: the exponent is 3, so three fours are multiplied, 4 × 4 × 4. Then multiply two at a time: 4 × 4 = 16, and 16 × 4 = 64. The common wrong answer here is 12, which comes from multiplying the base by the exponent, 4 × 3. Writing the three fours down is what prevents it, because 4 × 4 × 4 does not look like 4 × 3 once it is on paper.

Example 2 · A power inside a longer expression

Find 5 + 2 × 32. Exponents come before multiplying: 32 = 9, and the expression is now 5 + 2 × 9. Then multiplying before adding: 2 × 9 = 18, so 5 + 18 = 23. Two wrong routes are worth seeing. Adding first gives (5 + 2) × 9 = 63. Doing the multiplication before the exponent gives (5 + 6)2 = 121. Three routes, three answers, and only the order from Quiz 6 gives the one the writing actually says.

Example 3 · The locust ladder, one rung at a time

Each generation is about 20 times the one before. Start at 1, and multiply by 20 for each generation that passes.

GenerationsWritten as a powerThe multiplicationTimes bigger
12012020
220220 × 20400
320320 × 20 × 208,000

Each row multiplies the row above it by 20, and the exponent goes up by one. That is worth noticing now, because it is the whole content of a rule this leg reaches later: multiplying by another copy of the base adds one to the exponent. Check the last row by hand: 20 × 20 = 400, and 400 × 20 = 8,000.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. Before computing any power, write its factors out.

  1. Write 7 × 7 × 7 × 7 using an exponent, then find its value.
  2. Which is larger, 25 or 52? Give the value of each.
  3. Denise worked out 63 and wrote 18. Which statement names what she did?
    • A) She multiplied the base by the exponent, 6 × 3, instead of multiplying three sixes together.
    • B) She added the base and the exponent, 6 + 3, and then doubled.
    • C) She used the wrong base: the answer to 36 is 18.
    • D) Nothing: 18 is correct, because 63 means six multiplied by three.
  4. List every perfect square that falls between 30 and 90.
  5. Write 106 out in full as a number, and say how many zeros it has.
  6. Find the value of 3 + 4 × 22.
A detail from the 1859 engraving: three winged locusts in flight, drawn large, with more of the swarm behind them

ReadingA reading rest stop, for the stubborn ones.

The twenty, four hundred and eight thousand in this quiz are not ours. They are published by the Food and Agriculture Organization of the United Nations, which tracks desert locusts and warns the countries in their path. Scroll to the short section headed Key Facts, about halfway down: it gives the life cycle in weeks, the three increases by generation, and the figure for how much a swarm eats in a day. Every number in this quiz's locust problems came from those few lines, and they can be checked against the arithmetic you have just done.

The picture above is a detail of the 1859 engraving at the top of this page, not a photograph from that organization.

  1. A swarm grows about 20 times larger with each generation. After three generations, how many times its starting size is it? Write the answer as a power and as a number.
  2. The energy a moving body carries grows with the square of its speed. A falcon doubles its diving speed. Its energy of motion is multiplied by which of these?
    • A) 2, because the speed doubled and the energy doubles with it.
    • B) 4, because doubling the speed multiplies the energy by 22.
    • C) 8, because doubling the speed multiplies the energy by 23.
    • D) 16, because doubling the speed multiplies the energy by 24.
  3. Find both: (−5)2 and −52.
  4. A storage crate is a cube measuring 3 feet along every edge. Give the area of one of its square faces, and the number of one-foot cubes that would fill it. Write each as a power and as a number.
Answer Key

Check your work

Open the key: after you've finished all ten
1 74, which is 2,401
Four sevens are multiplied, so the exponent is 4 and the base is 7. Then 7 × 7 = 49, 49 × 7 = 343, and 343 × 7 = 2,401. Multiplying in pairs is quicker: 49 × 49 = 2,401, the same answer by Quiz 4's method. If you wrote 28, that is 7 × 4, the base multiplied by the exponent.
2 25 is larger: 32 against 25
25 is five twos multiplied: 2, 4, 8, 16, 32. 52 is two fives multiplied: 25. Swapping the base and the exponent gives a different number, and usually a very different one. That is worth holding on to, because the two are written with the same digits and read as opposites.
3 A.
63 is 6 × 6 × 6 = 216, and Denise's 18 is 6 × 3. It is her habit exactly: an old rule carried into a room where it does not hold, the way she took the smaller digit from the larger in Quiz 3, right-justified decimals in Quiz 14, dropped the sign in Quiz 21, and added where the relationship multiplied in Quiz 24. B) describes arithmetic nobody did: 6 + 3 doubled is also 18, which is why the option is here, and a coincidence is not a diagnosis. C) is wrong twice over, since 36 is 729. D) states her error as though it were the rule.
4 36, 49, 64 and 81
These are 62, 72, 82 and 92. The one below is 25, which is under 30, and the one above is 100, which is over 90. Read them off the gold diagonal rather than testing numbers one at a time. If you included 25 or 100, check the word between: the question asks for squares that fall inside those two ends.
5 1,000,000, with six zeros
The exponent on a power of ten counts the zeros after the 1, so 106 has six of them: one million. The reason is the place-value slide of Quiz 15, each factor of ten moving every digit one place to the left. If you wrote seven digits' worth of zeros, count them again against the exponent; the digit 1 is not one of them.
6 19
The exponent first: 22 = 4. Then the multiplication: 4 × 4 = 16. Then the addition: 3 + 16 = 19. If you wrote 28, you added 3 + 4 before multiplying, which is (3 + 4) × 4. If you wrote 121, you worked the whole left side and then squared it, (3 + 4 × 2)2. Quiz 6's order settles all three: parentheses, exponents, multiply and divide, add and subtract.
7 203, which is 8,000 times
One generation gives 201 = 20 times, two give 202 = 400 times, three give 203 = 8,000 times. Check it by hand: 20 × 20 = 400, then 400 × 20 = 8,000. If you wrote 60, that is 20 × 3, the same error the key names in question 3. The exponent counts the generations; the base is what one generation does.
8 B.
The word square in the question is the instruction: the multiplier is the speed change squared, and the speed change is 2, so the energy is multiplied by 22 = 4. A) treats the growth as though it kept step with the speed, which is what square rules out. C) and D) use 23 and 24, which would be right if the energy grew with the cube or the fourth power of the speed, and it does not. This is the same shape as the square and the cube in the Guide: which exponent applies is a fact about the world, and the arithmetic follows it.
9 (−5)2 = 25 and −52 = −25
In the first, the parentheses make −5 the base, so it is (−5) × (−5), and Quiz 22's rule makes that positive 25. In the second there are no parentheses, so the exponent belongs to the 5 alone: work 5 × 5 = 25 first, and the minus sign in front makes it −25. If you gave 25 for both, the parentheses were read as decoration. They are the whole difference.
10 One face: 32 = 9 square feet. The crate holds 33 = 27 one-foot cubes.
A face is a square 3 feet on each side, so its area is 3 × 3 = 9 square feet, which is Quiz 27's area. The whole crate is that face repeated through 3 feet of depth, so 9 × 3 = 27, or 3 × 3 × 3 = 33. Nine and twenty-seven come from the same number 3, and the exponent is all that separates them. If you wrote 27 for the face, you cubed where the question asked for a flat area; a face has two directions, not three.

Reading your results

QuestionsThe skill they testIf they gave trouble
1, 2, 7What the exponent countsWrite the factors out every time. The exponent says how many copies of the base to multiply, and never what to multiply the base by.
3Naming the errorBase times exponent is the one error to watch for in yourself. Writing the factors out is what catches it.
4, 5Perfect squares and powers of tenLearn the gold diagonal through 144 by sight. On a power of ten, the exponent counts the zeros.
6, 9Exponents among other operationsExponents come after parentheses and before multiplying. Parentheses decide whether a minus sign is inside the base.
8, 10Squares and cubes in the worldTwo directions square, three directions cube. Which exponent applies is a fact about the thing being measured.

Eight or more right: the reading is automatic, which is what this topic needed, and Quiz 30 turns every one of these powers around and asks what was squared or cubed to get there. 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: every time a raised number appears, write the multiplication out in full before doing anything else. Then come back to these same ten.

Where this goes

Next in Creatures in Flight: Quiz 30 — Square and Cube Roots. Every power in this quiz gets run backward. If 32 is 9, then the side of a square holding 9 is 3, and that is a square root; if 33 is 27, the edge of the cube is a cube root. Galileo drew the reason it matters in 1638, and a bone is what he drew.

The Company · An Interlude

Eight thousand times

The full 1859 engraving: locusts fill the sky and cover the ground in the foreground, while two men swing flails at them in front of a farmhouse
The engraving whole, as it was printed. Emil Schmidt, “A swarm of locusts,” in the German weekly Die Gartenlaube, 1859. The insects at the front are drawn nearly life size, which is what makes the smudge behind them legible as more of the same.

The men in this picture are doing what could be done in 1859, which was to stand in the field and swing. That is not foolish. A flail kills locusts. The difficulty is in the arithmetic, and the artist has drawn the arithmetic without naming it: three insects at the front, then a dozen, then a texture, then a sky that is no longer sky.

A detail from the engraving: a man in a cap, seen from behind, swinging a flail over his shoulder toward the swarm, with a farmhouse behind him
The same man, closer. Behind him the farmhouse; in front of him, according to the figures now published, as many as eighty million insects in a square kilometer.

The Food and Agriculture Organization of the United Nations has counted what these men could only swing at. A square kilometer of desert locust swarm can hold up to eighty million adults, and can eat in one day what thirty-five thousand people eat. A generation takes about three months, and each one is about twenty times the last, which is the twenty, four hundred and eight thousand of this quiz.

Notice what the exponent does for anybody trying to think about this. Eight thousand times is not a number a person can picture, and writing it as 203 does not make it picturable. What the power gives is something else: it says where the number came from. Three generations, twenty times each. Written that way the figure can be checked, argued with, and acted on before the third generation arrives rather than after it, which is the whole purpose of a warning.

Prints only this section, with the pictures at full size.