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A cube whose three dimensions are each one decimeter contains a liter.

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A kilogram is the weight of a liter of pure water at its maximum density. It is sometimes called a kilo.

ORAL EXERCISES

473. 1. How many cubic centimeters of water will weigh one kilogram? One gram?

2. How many cubic meters of water will weigh a metric ton?

3. What is the weight of a centiliter of metal that weighs 7 times as much as an equal quantity of water?

WRITTEN EXERCISES

474. 1. Find the number of liters in a rectangular tank 8 m. by 8 m. by 16 m.

Compare with Exercise 2, Art. 299. Which is simpler?

2. If wheat is .77 as heavy as water, find the number of kilograms of wheat a rectangular bin 5 m. long, 4 m. wide, and 2 m. deep will hold.

3. Find the capacity in hektoliters and the weight in metric tons of the water in a reservoir 40 m.

wide, and 52 dm. deep.

long, 26.6 m.

475. The pupil can transfer from units of one system to those of the other by using the following:

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For approximate estimates we may allow 1.1 yd. to 1 m., .6 mi. to 1 Km., and 1 qt. to 1 liter.

Use these values in oral exercises.

ORAL EXERCISES

476. 1. I bought 8 meters of silk for a skirt in the Palais Royal of Paris. How many yards did I buy?

2. Eleven yards will make a silk dress. How much must I buy in a Parisian shop?

3. How many liters of milk would supply for one week a family which uses a half gallon daily?

4. If two cities are 150 kilometers apart, what is the distance between them in miles?

WRITTEN EXERCISES

477. 1. The distance from one city to another is 420 Km. Express this distance in miles.

2. The Eiffel Tower in Paris is 300 m. high. How many feet high is it?

3. How many pounds will a person weigh who weighs 85 kilograms?

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POWERS AND SQUARE ROOT

478. 1. What is the product of 2 multiplied by 2, or used twice as a factor?

2. How many times must 3 be used as a factor to make 9? To make 27? (3 x 3 = 9; 3 x 3 x 3 = 27.) 3. If 5 × 5 × 5 = 125, how many factors of 125 are shown? How do they compare with each other?

479. The product of equal factors is called a Power. 480. The product of two equal factors is called the second power. Thus, since 3 x 3 = 9, 9 is the second power of 3.

The second power of a number is also called its square, because the area of a square is expressed by the product of two equal numbers that measure its length and breadth. (Art. 162.)

1. What is the area of a square 12 inches on a side? What is it called? (Art. 160.)

2. What is the second power or square of 12?

3. What is the second power or square of 7? Of 8? Of 9? Of 11?

481. The product of three equal factors is called the third power. Thus, since 5 x 5 x 5 = 125, 125 is the third power of three.

The third power of a number is also called its cube, because the volume of a cube is the product of the three equal numbers that measure its length, breadth, and thickness. (Art. 188.)

1. What is the volume of a cube 12 inches each way, and what is it called? (Art. 185.)`

2. What is the third power or cube of 12?

3. What is the third power or cube of 4? Of 6? 482. To show how many times a factor enters into a power, a small figure is placed to the right of the factor and slightly higher. Thus, 22 is read "2 square" or "2 to the second power," and means 2 × 2; 23 is read "2 cube,” or “2 to the third power," and means 2 × 2 × 2. Finding any power of a number is merely a case of multiplication, viz., finding the product of equal factors. 1. Write 3 × 3 as a power, and give its value.

3 x3=329.

Write the following as powers, and give their values:

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483. Since 4340+ 3, the square of 43 may be found

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That is, 432 (40 + 3)2 = 402 + 2 (40 × 3) + 32 = 1600 +

240 +9

=

= 1849.

Representing the number of tens by t and the number of ones by o, we may write

(t+o)2=t2+2txo+02.

Hence, the square of a number consisting of tens and ones equals the square of the tens, plus twice the tens multiplied by the ones, plus the square of the ones.

Note that in (a) on the preceding page the parts of the square are clearly shown, while in (b) and in the final result, 1849, they are not seen. Square the following, and point out the several parts of each square as in (a):

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484. 1. Separate 9 into two equal factors. Name one of them.

2. Separate each of the following into two equal factors: 16, 25, 36, 49.

485. A number that can be separated into 2 equal factors is called a perfect square. One of the two equal factors into which a number can be separated is called the Square Root of the number.

The square root of a number is indicated by the symbol Thus, √25 calls for the separation of 25 into two equal factors, or for the square root of 25.

486. Finding the square root of a number is the reverse of finding the square of a number. Just as finding a square is a special case in multiplication, finding a square root is a special case in division.

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