Before you begin
Quiz 29 built powers going out. This one brings them back. A root asks the question a power answers in reverse: not what is 3 squared, but what was squared to get 9. Every fact learned last time is worth double here, because a table of squares read left to right gives powers and read right to left gives roots. There is no new arithmetic to learn. There is a new question to hear, and a new symbol to write.
The symbol, and how to write it
The mark √ is called a radical sign. Whatever sits under its bar is the number you are asking about. So √ is read “the square root of 9,” and it means: the number that, multiplied by itself, gives 9. That number is 3, because 3 × 3 = 9. We write √ = 3.
A small 3 tucked into the notch, 3√, changes the question to a cube root. 3√ asks for the number that, multiplied by itself three times, gives 27. That number is 3 again, because 3 × 3 × 3 = 27. We write 3√ = 3.
What a root is, in a picture
Quiz 29 said a square 3 on a side holds 32 = 9 unit squares, and a cube 3 on an edge holds 33 = 27 unit cubes. Roots ask those two questions from the other end. Given a square of area 49, how long is its side? Given a cube of volume 27, how long is its edge?
Reading the diagonal backward
The perfect squares of Quiz 29 are the whole of square roots through 144. Read the left column against the middle one and you have powers; read the middle against the right and you have roots. It is one list, used in two directions, and it is worth knowing by sight.
Open the two lists: squares and their roots, cubes and theirs
| Number | Its square | The square root, backward |
|---|---|---|
| 1 | 1 | √ = 1 |
| 2 | 4 | √ = 2 |
| 3 | 9 | √ = 3 |
| 4 | 16 | √ = 4 |
| 5 | 25 | √ = 5 |
| 6 | 36 | √ = 6 |
| 7 | 49 | √ = 7 |
| 8 | 64 | √ = 8 |
| 9 | 81 | √ = 9 |
| 10 | 100 | √ = 10 |
| 11 | 121 | √ = 11 |
| 12 | 144 | √ = 12 |
The chart is wider than your screen: slide it sideways with your finger, or turn your phone.
| Number | Its cube | The cube root, backward |
|---|---|---|
| 1 | 1 | 3√ = 1 |
| 2 | 8 | 3√ = 2 |
| 3 | 27 | 3√ = 3 |
| 4 | 64 | 3√ = 4 |
| 5 | 125 | 3√ = 5 |
| 6 | 216 | 3√ = 6 |
| 7 | 343 | 3√ = 7 |
| 8 | 512 | 3√ = 8 |
| 9 | 729 | 3√ = 9 |
| 10 | 1000 | 3√ = 10 |
Cubes climb much faster than squares, so far fewer of them are worth memorizing. Through 10 is plenty, and the test rarely asks past 1,000.
When the root is not a whole number
Most numbers are not perfect squares, and their roots are not whole. √ is not a tidy number, but you can still say a great deal about it without a calculator, and the GED often asks for exactly that much. 50 sits between the perfect squares 49 and 64. So √ sits between 7 and 8, and because 50 is very close to 49 and far from 64, the root is very close to 7.
That is Quiz 2's estimate, still standing guard. Naming the two whole numbers a root falls between will answer many test questions outright, and on the rest it tells you at once whether the calculator's answer is believable.
What roots are for: getting a size back
Powers take a length and give you an area or a weight. Roots run that backward, and the reason it matters is that weights are what get measured and lengths are what get built.
In 1638 Galileo published a book with the drawing at the top of this page in it. His point was that a bone cannot simply be scaled up. Make an animal 3 times longer in every direction and its weight goes up by 33, which is 27 times, while the width of its bones goes up by only 32, which is 9 times. So the bone has to be thickened out of proportion, and he drew what that looks like. Three hundred years later the biologist J. B. S. Haldane worked the same argument through the animals, and named Galileo while doing it.
Haldane put the flight case in numbers that this quiz can check. Take two flying machines of the same shape, one larger. If the larger one weighs 64 times as much, then since weight follows the cube of the length, its length is 3√ = 4 times greater. And the least speed it needs to stay in the air follows the square root of its length, so that speed is √ = 2 times greater. One object, both roots: a cube root to get from weight back to length, and a square root to get from length to speed. This is also why the largest birds soar rather than flap, and why there is a size past which nothing flies at all.
Three to study before you start
Find √. Ask the question in words: what number, multiplied by itself, gives 121? Then hunt on the squares list: 10 × 10 = 100, too small; 11 × 11 = 121, exactly. So √ = 11. Check by squaring back: 11 × 11 = 121, which is where we started. That check costs one multiplication and catches every wrong answer, including the halving error, since 121 ÷ 2 is 60.5 and 60.5 × 60.5 is nothing like 121.
Find √. The bar covers both numbers, so add first: 36 + 64 = 100. Then take the root: √ = 10. Now the other reading, so you can see the size of the difference: √ + √ is 6 + 8 = 14. Ten and fourteen, from the same two numbers. The bar told you which one was being asked for, and it is the only thing that did.
A flying machine is built to the same shape as a smaller one but weighs 64 times as much. How much longer is it, and how much faster must it fly to stay up?
Weight follows the cube of the length, so to get the length back, undo a cube: 3√ = 4, because 4 × 4 × 4 = 64. It is 4 times longer. The least flying speed follows the square root of the length, so take a square root of that 4: √ = 2. It must fly 2 times as fast. Notice the order of the questions. The weight was known and the length was wanted, so the cube root came first; the length then fed the square root. Roots are how a measurement you can take gives you back a measurement you cannot.
Now you
Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. After every root you find, square it or cube it back and see whether you land where you started.
- Find √ and √.
- Find 3√ and 3√.
- Marcus worked out √ and wrote 18. Which statement names what he did?
- A) He subtracted 2 from 36 and then halved the result.
- B) He divided by 2 instead of asking what number times itself gives 36.
- C) He found the cube root instead of the square root.
- D) Nothing: 18 is correct, because a square root is half of the number.
- A square patio covers 144 square feet. How long is each side?
- Find both, and say why they differ: √ and √ + √.
- √ falls between which two whole numbers, and which of the two is it nearer? Say how you know.
ReadingA reading rest stop, for the stubborn ones.
The flying machine in Example 3 comes from a short essay of 1926, On Being the Right Size by J. B. S. Haldane, printed here in full and free to read. It is about six pages. Two places to land: the second paragraph, on the sixty-foot giants whose thigh bones would break at every step, and the paragraph beginning “Exactly the same difficulties attach to flying,” about two thirds of the way down, which is where the square root and the 64 and the 128 come from. Everything in between is the same argument run through mice, insects and eyes.
The picture above is Galileo's woodcut of 1638, shown at the top of this page. It is not from that essay, though the essay credits Galileo for the argument.
- A cube-shaped crate holds 64 cubic feet. How long is each edge?
- Two aircraft are built to the same shape, and the larger weighs 64 times as much as the smaller. Weight follows the cube of the length. How many times longer is the larger one?
- A) 8 times, because √ = 8.
- B) 4 times, because 3√ = 4.
- C) 64 times, because the weight and the length grow together.
- D) 32 times, because 64 ÷ 2 = 32.
- Find the value of 3 + 2 × √.
- A cube-shaped tank holds 1,000 cubic inches. Find the length of one edge, and then the area of one of its square faces.
Check your work
Open the key: after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1, 2 | Roots of perfect squares and cubes | Learn the squares to 144 and the cubes to 1,000 by sight, then read them backward. Square or cube your answer to check it. |
| 3 | Naming the error | A square root is never half. Multiply the answer by itself; if it does not return the original number, the answer is wrong. |
| 5, 9 | How far the bar reaches | Everything under the bar is worked out first, exactly as in parentheses. Draw your bars carefully. |
| 6 | Estimating a root | Name the perfect squares on either side. Their roots are the two whole numbers the answer falls between. |
| 4, 7, 8, 10 | Which root the shape asks for | Two directions take a square root, three take a cube root. Areas give back sides; volumes and weights give back lengths. |
Eight or more right: powers now run in both directions, which is what Quiz 31 needs, since it puts them together and takes them apart. 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: after every root, multiply the answer by itself the right number of times and see whether you land back where you started. Then come back to these same ten.
Next in Creatures in Flight: Quiz 31 — Rules of Exponents. Powers get multiplied, divided, and raised to further powers, and each rule turns out to be counting factors rather than memorizing a line. It is also where 200 gets an answer, by walking the locust ladder of Quiz 29 back down one rung at a time. A flock of starlings holds the pictures.
The bone that had to be redrawn
Galileo was seventy-four when this page was printed, under house arrest, and not permitted to publish in his own country; the book was carried out and printed in the Netherlands. Much of it is about falling bodies. This page is about bones, and it holds the drawing that the mathematics in this quiz is for.
His argument is the one this quiz keeps running backward. Make an animal three times longer in every direction and it weighs 27 times as much, because weight follows three directions at once. But the bone that has to carry that weight is only 9 times wider across, because a cross section has two directions. So a bone that is merely scaled up will snap. To hold, it has to be thickened out of all proportion, and the lower figure is Galileo drawing what that would look like. He adds that a very large animal built to a small animal's plan would be crushed by its own weight.
In 1926 the biologist J. B. S. Haldane took that argument through the animals in a short essay, and complained that people had gone on ignoring it for three hundred years. A giant ten times a man's height would weigh a thousand times as much while his bone cross sections grew only a hundredfold, so every square inch of him would carry ten times the load: his thighs would break the moment he stood up. Downward it runs the other way, and small things are nearly safe from falling. Drop a mouse down a mine shaft, he wrote, and it walks away; a rat is killed, a man is broken, and a horse splashes.
Flight is where the roots come in. Haldane wrote that the least speed a flying machine needs to stay up follows the square root of its length. Four times the length, twice the speed. And the power it takes climbs faster than the weight, so an aircraft 64 times heavier needs 128 times the horsepower. Run that up through the birds and the limit arrives quickly. A bird as large as we might imagine would need a breastbone standing feet out from its chest to hold the muscle, so the big ones stopped flapping and learned to ride rising air instead. Haldane's summary of what that leaves us is worth keeping: eagles are not the size of tigers, and there is a reason.
The arithmetic in all of it is small. Weight to length is a cube root, length to speed is a square root, and both are the questions of this quiz asked about something with wings.
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