Creatures in Flight · Quiz 31a · Fractional Exponents

The Guide

Before you begin

This quiz is about an exponent that is a fraction, such as 4^1/2 or 4^3/2, and what it asks for. It asks for two things, one from each half of the fraction: the bottom is a root, and the top is a power. 4^1/2 is the square root of 4, which is 2, and nothing more, because the top is 1. 4^3/2 is that same root raised to the third power, 2 × 2 × 2 = 8. The reason comes from the rules of exponents, which this Guide lays out as it goes. If you did Quiz 30 and Quiz 31, you have the two pieces already: the radical sign for writing roots, and the rules with the names on them, ADD, SUBTRACT and MULTIPLY. If not, everything you need is below. Two names are still missing from the chart of those rules, ROOT and DISTRIBUTE, and they arrive here. The dragonfly at the top of the page comes with a number that shows why anyone would want to write a root that way.

What 4^1/2 has to mean

The rules do not care what kind of number the exponent is. MULTIPLY says that (4^1/2)² = 4^{1/2 × 2} = 4¹ = 4. So whatever 4^1/2 is, squaring it gives 4. Quiz 30 has a name for the number whose square is 4: the square root, √4 = 2. So 4^1/2 = 2, and for the same reason 9^1/2 = 3 and 25^1/2 = 5. The same argument runs with 1/3. (8^1/3)³ = 8^{1/3 × 3} = 8¹ = 8, so 8^1/3 is the number whose cube is 8, which is 3√8 = 2. And with 1/4: (16^1/4)⁴ = 16, so 16^1/4 is the number that, used as a factor four times, gives 16. That is 2, because 2 × 2 × 2 × 2 = 16.

Quiz 30 wrote roots with the radical sign: √4 means the square root of 4, the number that multiplied by itself gives 4. A small 3 tucked into the notch of the sign, 3√8, asks for a cube root, and a small 4, 4√16, for a fourth root. The bottom of the fraction is that small number in the notch. An exponent of 1/2 is a square root, 1/3 is a cube root, 1/4 is a fourth root. This is ROOT, the sixth name on the Exponent Machine’s chart, and it is MULTIPLY read backward.

Quiz 30’s two pictures with the new writing on them. The area 4 gives back its side as 4^1/2 = 2, and the volume 8 gives back its edge as 8^1/3 = 2. The number of directions in the shape is the bottom of the fraction.

What the fraction is for. Once a root is written as an exponent, the rules from Quiz 31 work on it without being told anything new. A root and a power can sit in one exponent, as 4^3/2 does below; two roots taken one after the other can be joined into one, as they are for the flying machine near the end of this Guide; and a root of a product can be split by DISTRIBUTE. The radical sign can do none of that by itself. Written as a fraction, a root obeys the same rules as any other exponent, and that is what the fraction is for.

The top of the fraction: a power on top of a root

Take 4^3/2. MULTIPLY reads 3/2 as 1/2 × 3, so 4^3/2 = (4^1/2)³: take the root, then cube it. √4 = 2, and 2³ = 8. Or read 3/2 as 3 × 1/2, so 4^3/2 = (4³)^1/2: cube first, then take the root. 4³ = 64, and √64 = 8. Both roads reach 8, because 1/2 × 3 and 3 × 1/2 are the same number, and MULTIPLY does not care in which order it is written.

Root first keeps the numbers small: 2, then 8. Power first passes through 64 on the way to the same 8. Both are right; one is easier to do in your head.

Bottom is the root, top is the power, and take the root first, because the numbers stay small. 8^2/3: 3√8 = 2, then 2² = 4. 27^2/3: 3√27 = 3, then 3² = 9. 16^3/4: 16^1/4 = 2, then 2³ = 8. Try 16^3/4 the other way to see what you are spared: 16³ = 4,096, and then you would need the fourth root of 4,096, which is 8, but few people can see that by looking.

Brian stops at the root. He works out 16^3/4 and writes 2. He took the fourth root of 16, which is 2, and stopped: digits right, one step missing, which is his habit exactly, the way he dropped the slide in Quiz 4, dropped a decimal place in Quiz 15, answered the discount when the question asked the price in Quiz 19, and applied a length factor once where an area needed it twice in Quiz 26. Here the missing step is the 3 on top of the fraction. 2 is 16^1/4, and 3/4 is three of those: (16^1/4)³ = 2³ = 8. The check that catches it: raise the answer to the bottom of the fraction and see whether you get the top’s power of 16. 8⁴ = 4,096 = 16³, so 8 is right; 2⁴ = 16 = 16¹, so 2 answered a different question.

DISTRIBUTE: an exponent outside parentheses

Take (4 × 9)^1/2. One road is to work the inside first: 4 × 9 = 36, and 36^1/2 = 6. The other road hands the exponent to each factor inside: 4^1/2 × 9^1/2 = 2 × 3 = 6. Both roads reach 6, and the reason is a count of factors again. (2 × 3)² means (2 × 3) × (2 × 3), which is two 2s and two 3s, which is 2² × 3². An exponent outside parentheses is given to every factor inside them. That is DISTRIBUTE, the seventh name on the chart, and it works for fractions in the exponent because it works for every exponent.

The expression	Inside first	Distribute first
(4 × 25)^1/2	100^1/2 = 10	4^1/2 × 25^1/2 = 2 × 5 = 10
(8 × 27)^1/3	216^1/3 = 6	8^1/3 × 27^1/3 = 2 × 3 = 6
(1625)^1/2	1625 is 0.64, and 0.64^1/2 = 0.8	45, which is 0.8
(2 × 3)²	6² = 36	2² × 3² = 4 × 9 = 36

The same shape four times, and the two columns agree in every row. DISTRIBUTE crosses a multiplication sign or a fraction bar. It never crosses a plus or a minus. (9 + 16)^1/2 is 25^1/2 = 5, and it is not 9^1/2 + 16^1/2, which is 3 + 4 = 7. That is Quiz 30’s rule that the bar of the radical sign is a fence, now with a name on it: DISTRIBUTE stops at a plus or a minus sign.

Denise distributes over the plus sign. She writes (9 + 16)^1/2 = 3 + 4 = 7. It is her habit: an old habit carried where it does not hold, the way she took the smaller digit from the larger to dodge a trade in Quiz 3, right-justified decimals in Quiz 14, dropped a sign in Quiz 21, added where the relationship multiplied in Quiz 24, and again in Quiz 29. The habit here is a good one in its own room: 2 × (3 + 4) really is 2 × 3 + 2 × 4. But an exponent is not a multiplier standing outside the parentheses; it is a count of factors of what is inside, and 9 + 16 is one number, 25. The check: 7² = 49, and 49 is not 25, while 5² = 25.

Where fractions in the exponent come from: size

Quiz 30 built a flying machine 64 times as heavy as a smaller one of the same shape. Weight follows the cube of the length, so the length is the cube root of the weight: 64^1/3 = 4 times as long. The least flying speed follows the square root of the length: 4^1/2 = 2 times as fast. Two roots, one after the other. Written as exponents they join into one by MULTIPLY: the speed is the weight to the 1/3 × 1/2, which is the weight to the 1/6. Check: 64^1/6 is the number that, used as a factor six times, gives 64, and 2 × 2 × 2 × 2 × 2 × 2 = 64, so it is 2. The radical sign could not have written that sentence; the fraction did it in one step.

Now the dragonfly. The one at the top of this page is a small skimmer. A big dragonfly today spans about 7 centimeters across its wings. Three hundred million years ago, before the dinosaurs, the same body plan was flying with wings 70 centimeters across: the griffinflies, giant relatives of dragonflies, whose fossils sit in museums in Paris and London. Ten times the length. Suppose for a moment that the giant was the same shape as a dragonfly today, only larger, which a curator at the Natural History Museum in London says is nearly true. Then its wing area was 10² = 100 times as great, and its weight 10³ = 1,000 times as great. That is Quiz 29’s square and cube. Read the ladder the other way and the fractions appear: the length is the weight to the 1/3, since 1,000^1/3 = 10, and the wing area is the weight to the 2/3, since 1,000^2/3 = (1,000^1/3)² = 10² = 100. Wing area is the two-thirds power of weight. Turned around, weight is the three-halves power of wing area: 100^3/2 = (100^1/2)³ = 10³ = 1,000.

The same three numbers, 10, 100 and 1,000, written twice: going up as powers of 10, and coming down from the weight as fractions. The 2/3 and the 1/3 are not new facts; they are the square and the cube read from the other end.

Whether the giant really weighed a thousand times a dragonfly is a claim the fossils can test, and the gallery at the foot of the page tests it. The arithmetic of the fraction does not depend on the answer.

The three-quarter rule. One more fraction lives in the world of animals, and it is measured rather than derived. In 1932 Max Kleiber compared the energy animals burn at rest, from mice to cattle, and found that it does not double when the weight doubles. It grows more slowly, close to the three-quarter power of the weight. An animal 16 times as heavy burns about 16^3/4 = (16^1/4)³ = 2³ = 8 times the energy, not 16 times. Biologists still argue about why the number is three quarters. Nobody argues about the arithmetic: the bottom of the fraction is a fourth root, the top is a cube, and the root comes first.

The chart, complete

Open the chart: all eight names, with a check for each

Name	When	The move	Check with numbers
ADD	multiplying powers, same base	2³ × 2⁴ = 2⁷	8 × 16 = 128
SUBTRACT	dividing powers, same base	2⁵ ÷ 2² = 2³	32 ÷ 4 = 8
MULTIPLY	a power of a power	(2³)² = 2⁶	8 × 8 = 64
FLIP	negative exponents	2⁻³ = 1 over 2³	1 ÷ 8 = 0.125
DISTRIBUTE	an exponent outside parentheses	(4 × 9)^1/2 = 4^1/2 × 9^1/2	6 = 2 × 3
ONE	the zero exponent	20⁰ = 1	20 ÷ 20 = 1
ROOT	fraction exponents: the bottom is a root, the top a power	4^3/2 = (√4)³	2³ = 8, and √64 = 8
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.

The printable one-page version, the Rules of Exponents chart, is in the same folder as this quiz: exponent-rules-chart.html. With this quiz, every name on it has been earned.

Worked Examples

Three to study before you start

Example 1 · Three roots, written as fractions

Find 49^1/2, 125^1/3 and 81^1/4. Read the bottom of each fraction as the root: 49^1/2 asks what number, squared, gives 49; that is 7, since 7 × 7 = 49. 125^1/3 asks what number, cubed, gives 125; that is 5, since 5 × 5 × 5 = 125. 81^1/4 asks what number, used as a factor four times, gives 81; that is 3, since 3 × 3 × 3 × 3 = 81. So the three answers are 7, 5 and 3. Check by MULTIPLY: (49^1/2)² = 49¹, and 7² = 49, which is where we started.

Example 2 · A power on top of a root, both roads

Find 8^2/3. Root first: the bottom is 3, so take the cube root, 3√8 = 2, and then the top is 2, so square it: 2² = 4. Power first, to see that it agrees: 8² = 64, and then the cube root, 3√64 = 4, because 4 × 4 × 4 = 64. Same answer, and the first road never left single digits. Now 27^2/3 the same way: 3√27 = 3, and 3² = 9. If you ever forget which number is the root, write the fraction as 1/3 × 2 and let MULTIPLY read it: a third of the way is the root, and the 2 is the power.

Example 3 · DISTRIBUTE, and the sign it will not cross

Find (9 × 16)^1/2, and then (9 + 16)^1/2. The first has a multiplication inside, so DISTRIBUTE may be used: 9^1/2 × 16^1/2 = 3 × 4 = 12. Check by working the inside first: 9 × 16 = 144, and 144^1/2 = 12, from the squares list. The second has a plus sign inside, so DISTRIBUTE may not be used: work the inside first, 9 + 16 = 25, and 25^1/2 = 5. Handing the exponent across the plus sign would give 3 + 4 = 7, and 7² = 49, which is not 25. The two expressions look alike, and the sign inside the parentheses is what decides which rule applies.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. For every fraction in an exponent, say to yourself which number is the root and which is the power before you touch either.

Write 25^1/2 and 27^1/3 with radical signs, and find both.
Find 16^1/4 and 81^1/4, and say what the 4 in the exponent asks for.
Brian worked out 16^3/4 and wrote 2. Which statement names what he did?
- A) Nothing: 16^3/4 is 2, because a fraction in the exponent means a root and the root of 16 is 2.
- B) He took the cube root of 16, which is about 2.5, and rounded it down.
- C) He took the square root instead of the fourth root.
- D) He took the fourth root, which is 2, and stopped, leaving out the cube that the 3 on top asks for; the answer is 2³ = 8.
Find 4^3/2 by both roads, root first and power first, and show that they agree.
Find 8^2/3 and 27^2/3.
Find (4 × 25)^1/2 by both roads. Then find (9 + 16)^1/2, and say why the second one cannot be split the way the first one can.

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

Open the Exponent Machine, in the same folder as this quiz at thepeoplesshare.org/ged/math/exponent-machine.html, and tap the two tiles this quiz is about. ROOT derives the fraction exponent from MULTIPLY and works 4^3/2 in front of you; DISTRIBUTE hands an outside exponent to everything inside the parentheses, and its quiz mode sets a trap with (x · y)³, which looks like CAN’T COMBINE and is not.

ReadingOr read instead: Griffinflies: the earliest flying insects, a short page from the Natural History Museum in London. Land on the section headed The biggest dragonfly of them all, near the foot of the page, which gives Meganeura monyi’s weight as about that of a large apple; the gallery below does the arithmetic with it.

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

A griffinfly is 10 times as long as a dragonfly today. If it were the same shape, how many times the wing area would it have, and how many times the weight? Then write the wing area as a power of the weight, using a fraction in the exponent.
By Kleiber’s rule, an animal 16 times as heavy as another burns about 16^3/4 times the energy at rest. What number is that?
- A) 12, because 16 × 3/4 = 12.
- B) 4, because the square root of 16 is 4.
- C) 8, because 16^1/4 = 2 and 2³ = 8.
- D) 64, because 16³ is 4,096 and its square root is 64.
Find (2549)^1/2 and (8 × 27)^1/3.
In Quiz 30, a flying machine 64 times as heavy as another was 64^1/3 = 4 times as long, and had to fly 4^1/2 = 2 times as fast. Write the speed as a single power of the weight, with one fraction in the exponent, and check that it gives 2.

Answer Key

Check your work

Open the key: after you've finished all ten

1 √25 = 5 and 3√27 = 3

The bottom of 1/2 is a square root and the bottom of 1/3 is a cube root. 5 × 5 = 25, and 3 × 3 × 3 = 27. If you wrote 12.5 for the first, you multiplied 25 by 1/2; an exponent is never a multiplier, and the check shows it: 12.5 × 12.5 is 156.25, nowhere near 25.

2 16^1/4 = 2 and 81^1/4 = 3

The 4 asks for a fourth root: the number that, used as a factor four times, gives the base. 2 × 2 × 2 × 2 = 16 and 3 × 3 × 3 × 3 = 81. If you wrote 4 for the first, you took a square root, which is what a 2 on the bottom would have asked for; 4 × 4 × 4 × 4 is 256, not 16.

3 D.

16^3/4 is a fourth root, 2, and then a cube, 8. Brian did the root and stopped, which is his habit of one step missing, as in Quizzes 4, 15, 19 and 26. A) states his error as the rule, and the check refuses it: 2⁴ = 16, so 2 is 16^1/4, not 16^3/4. B) reads the 3 on top as the root, and it is the bottom of the fraction, 4, that names the root; besides, rounding is not a move in this arithmetic, and 16^1/4 is exactly 2 with nothing to round. C) would give 4, since √16 = 4, and he wrote 2. The right answer is 8, and 8⁴ = 4,096 = 16³ confirms it.

4 8, by both roads

Root first: √4 = 2, then 2³ = 8. Power first: 4³ = 64, then √64 = 8. They agree because 1/2 × 3 and 3 × 1/2 are the same exponent, 3/2, and MULTIPLY reads it either way. If your two roads gave different numbers, one of them slipped; the root-first road is the one to trust, since it never leaves small numbers.

5 8^2/3 = 4 and 27^2/3 = 9

Cube root first, then square. 3√8 = 2 and 2² = 4; 3√27 = 3 and 3² = 9. If you wrote 8 for the first, you squared 8 to 64 and then took a square root instead of a cube root: 64^1/2 is 8, but 64^1/3 is 4, and the bottom of the fraction says which.

6 (4 × 25)^1/2 = 10; (9 + 16)^1/2 = 5, and it cannot be split because the sign inside is a plus

Inside first: 4 × 25 = 100, and 100^1/2 = 10. DISTRIBUTE: 4^1/2 × 25^1/2 = 2 × 5 = 10. For the second, inside first is the only road: 9 + 16 = 25, and 25^1/2 = 5. DISTRIBUTE hands an exponent to each factor of a product; 9 + 16 is not a product of anything, it is the one number 25, so there is nothing to hand the exponent to. Handing it across the plus sign anyway gives 3 + 4 = 7, and 7² = 49 is not 25.

7 100 times the wing area, 1,000 times the weight; the wing area is the weight to the 2/3

Ten times the length in every direction gives 10² = 100 times the area and 10³ = 1,000 times the weight, which is the square-cube law from Quiz 29. Reading back from the weight: 1,000^1/3 = 10 is the length, and 1,000^2/3 = 10² = 100 is the wing area, so the wing area is the weight to the 2/3. If you wrote 1,000^3/2, you turned the fraction over, and 1,000^3/2 is 10^9/2, a five-digit number.

8 C.

Bottom is the root, top is the power: 16^1/4 = 2, then 2³ = 8. A) multiplies the base by the exponent, which is never what an exponent means. B) reads the 4 on the bottom as a square root and forgets the top. D) mistakes the fourth root for a square root on the power-first road: 4,096^1/2 is 64, but the bottom of the fraction asks for 4,096^1/4, which is 8. Kleiber’s rule is a measurement and not a law of arithmetic, but the arithmetic of 16^3/4 is not in doubt.

9 57 and 6

DISTRIBUTE crosses a fraction bar: (2549)^1/2 = 25^1/2 over 49^1/2 = 57. Check: 57 squared is 2549. And it crosses a multiplication sign: (8 × 27)^1/3 = 8^1/3 × 27^1/3 = 2 × 3 = 6. Check by the other road: 8 × 27 = 216, and 6 × 6 × 6 = 216.

10 64^1/6 = 2

The length is the weight to the 1/3, and the speed is the length to the 1/2, so the speed is the weight to the 1/3 × 1/2 = 1/6, by MULTIPLY. 64^1/6 is the number that, used as a factor six times, gives 64, and 2 × 2 × 2 × 2 × 2 × 2 = 64, so it is 2, which is the 2 Quiz 30 reached by two separate roots. If you wrote 64^5/6, you added the fractions; two roots taken one after the other are a power of a power, and MULTIPLY is the move.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2	ROOT: the bottom of the fraction	The bottom of the fraction is the small number in the notch of the radical sign. Check by raising your answer to that number; it should give the base back.
3	Naming the error	The top of the fraction is a power that still has to be applied after the root. Two numbers in the exponent, two steps.
4, 5	A power on top of a root	Root first, then power. Both roads agree, and the root-first road keeps the numbers small.
6, 9	DISTRIBUTE, and the sign it will not cross	An outside exponent goes to every factor inside, across × and across a fraction bar, never across + or −. Check any split by working the inside first.
7, 8, 10	Fractions in the exponent, in the world	Read the fraction as MULTIPLY: 2/3 is a cube root and then a square; 1/3 × 1/2 is one root after another, joined into 1/6. Write the two steps before doing either.

Eight or more right: the chart is complete and every name on it is yours, which is what Quiz 32 needs, since scientific notation is powers of ten with the exponents doing the work. 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 touching any fraction in an exponent, say aloud which number is the root and which is the power. Then come back to these same ten.

Where this goes

Next in Creatures in Flight: Quiz 32 — Scientific Notation. Very large and very small numbers, written as a number between 1 and 10 times a power of ten, so that Quiz 15’s powers of ten and Quiz 31’s negative exponents meet, and the exponent does the counting that the zeros used to do. The pictures stay with creatures in flight.

The Company · An Interlude

Ten times the wing

A dragonfly seen from directly above, perched on the tip of a stem, its four wings spread flat and marked with dark and amber patches, every wing divided into hundreds of small cells by a net of veins — A skimmer, one of the common families of dragonflies: *Palpopleura sexmaculata*, from directly above, in its own colors. Photograph by Mildeep, 2022, from Wikimedia Commons, under the Creative Commons Attribution-ShareAlike 4.0 license; shown here as taken. A copy recolored for this quiz’s masthead carries the same license.

The wings deserve a look before the numbers do. Each one is a sheet thinner than paper, held stiff by a net of veins, and the veins divide it into hundreds of small cells. The cells are what let a wing that weighs almost nothing take the strain of flight without tearing. A dragonfly’s four wings work on their own hinges, so it can hover, fly backward, and turn in its own length, which is how it catches other insects in the air.

A dragonfly perched on the tip of a dry stem, seen from the side, with amber and black patches across its four wings and a yellow and black banded body — A relative in Africa, *Palpopleura jucunda*, a female, on a dry stem. Photograph by JP Labuschagne, 2015, from Wikimedia Commons, under the Creative Commons Attribution-ShareAlike 4.0 license; shown as taken.

Three hundred million years ago, before there were dinosaurs, insects built on this plan flew with wings 70 centimeters across. They are called griffinflies, and they are not quite dragonflies, but the curator who looks after them at the Natural History Museum in London says the body plan has hardly changed in 350 million years; it has only become smaller. The first fossil, a single wing, was dug out of a coal mine at Commentry in France in 1880 and named Meganeura, which means large-veined, in 1885. The largest of all, Meganeuropsis permiana, from Kansas, spanned about 71 centimeters. Why they could be so large is still argued; the air then held more oxygen than it does now, and there were no birds or bats to catch them.

Now the arithmetic. A big dragonfly today spans about 7 centimeters and weighs about a gram. The same shape at 10 times the length would have 100 times the wing area and 1,000 times the weight: about a kilogram, a little more than two pounds. The museum’s estimate for Meganeura monyi is 100 to 150 grams, the weight of a large apple. So the same-shape model is wrong by somewhere between six and ten times, and that is the useful finding. The giant was not a dragonfly scaled up. It was built lighter for its length than a dragonfly is, with long narrow wings and a slender body, and the fossils are what tell you so. The fraction in the exponent, wing area as the weight to the 2/3, is the arithmetic of an animal that keeps its shape while it grows; when the shape changes, the fraction changes with it, and measuring the fraction is one of the ways biologists find out that it did.

A dragonfly perched on a twig, its wings marked with dark patches and glossy panels of amber, copper and violet, and its body banded red and blue; the photographer's mark, the letters KB, sits at the right edge — A third skimmer, species not identified, photographed in 2008. Photograph by K B, from Flickr by way of Wikimedia Commons, under the Creative Commons Attribution 2.0 license; shown as taken. The KB at the right edge is the photographer’s own mark and stays.

Kleiber’s three-quarter rule is the other fraction in this quiz, and it is the opposite kind of fact. Nobody derived it from a shape. Max Kleiber measured it in 1932, across animals from mice to cattle, and it has held up for nearly a century while the reasons for it are still argued over. An animal 16 times as heavy burns about 8 times the energy at rest, because 16^3/4 = 8, and the fraction is the whole of the claim. Two fractions, then: one that follows from the shape of a thing and can be worked out on paper, and one that only measurement could find. The arithmetic for both is the same: a root, and then a power.

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