SECTION XVII.

A. 1. A power of a number is the product obtained by taking the given number any number of times as a factor. It is sometimes defined as the product that would be obtained by multiplying the number once or more times by itself.

2. If a number is taken twice as a factor, it is said to be raised to the second power; if taken 3 times, to the third power, &c., &c. Thus, 9 = the second power of 3, because it is the product of 3 × 3, or 3 taken twice as a factor. 27= the third power of 3, because it is the product of 3 × 3 × 3, or of 3 threes multiplied together, or of 3 taken 3 times as a factor.

3. The second power of a number is often called its square, and the third power its cube.

4. The process of finding the second power of a number is called squaring the number, or raising it to the second power.

5. The process of finding the third power of a number is called cubing the number, or raising it to the third power.

6. The process of finding the fourth power of a number is called raising it to the fourth power,—of finding the fifth power is called raising it to the fifth power, &c.

7. The power to which a number is to be raised, is often indicated by a small figure placed to the right and a little above the number to be raised. The figure so placed is called an exponent.

power,
25.

i. l.,

8. 52 indicates that 5 is to be raised to the second taken twice as a factor. Therefore 52 = 5 X 5 9. 25 indicates that 2 is to be raised to the fifth power, i. e., taken 5 times as a factor. Therefore, 25 = 2 × 2 × 2 × 2 × 2 32.

10. 52 is read five square, or five second power.

11. 25 is read two fifth power.

12. The number thus repeated as a factor is called the root of the product obtained. Thus, 3 is the square root or second root of 9, because, taken as a factor twice, it will produce 9. It is the cube root or third root of 27, because, taken as a factor 3 times, it will produce 27. It is the fourth root of 81, because taken as a factor 4 times it will produce 81.

13. In like manner 2 is the square root or second root of 4, the cube root or third root of 8, the fourth root of 16, the fifth root of 32, the sixth root of 64, &c., &c.

B. 1. What is the square of 2? of 1? of 10? of 7? of 4? 2. What is the second power of 3? of 5? of 9? of 6? of 8? 3. What is the cube of 5?

Solution. The cube of 5 is the product of 5 taken as a factor 3 times, or of 5 × 5 × 5; but 5 × 5 = 25, and 25 × 5= 125. Therefore 53: = 125.

4. What is the cube of 7? of 3? of 10? of 2?

5. What is the third power of 8? of 4? of 1? of 6? of 9? 6. What is the value of 5'? of 25? of 73? of 34? of 45? of 35? 7. What is the square root of 81?

Ans. The square root of 81:

twice, or 9 X 9, equals 81.

= 9, because 9 taken as a factor

8. What is the square root of 64? of 1? of 4? of 25? 9. What is the square root of 49?

of 9? of 36? of 16? of 100?

10. What is the square root of 81? 11. What is the cube root of 8? of 729? of 1? of 343? of 64? 12. What is the cube root of 27? of 512? of 125? of 216? of 1000 ?

13. What is the fourth root of 81? of 256? of 16? of 625?

C. 1. How many square rods in a rectangular play-ground 9 rods long and 7 wide?

Solution. Since a space 1 rod long and 1 rod wide contains 1 square rod, a space 9 rods long and 1 rod wide must contain 9 square rods, and a space 9 rods long and 7 rods wide must contain 7 times 9 square rods, or 63 square rods. Therefore, a play-ground 9 rods long and 7 rods wide contains 63 square rods.

2. How many square rods in a field 40 rods long and 9 rods wide?

3. How many square feet in a platform 16 feet long and 9 feet wide? How many square yards?

4. How many square feet in a wall 98 feet long and 5 feet high? How many square yards?

5. How many square inches in a slate 10 inches long and 8

inches wide?

6. How many square yards in the floor of a room 6 yards long and 5 yards wide? How many square feet in the same floor?

7. How many square feet in a blackboard 16 feet long and 4 feet wide? How many square yards in the same board?

8. At 3 cents per square foot, what will it cost to paint a surface 16 feet long and 9 feet wide?

9. At 8 cents per square yard, what will it cost to plaster the ceiling of a school-room 14 yards long and 8 yards wide?

10. A teacher, wishing to obtain a blackboard 15 feet long and 6 feet wide, bought boards for the purpose, at 2 cents per square foot. He hired a carpenter to make it, paying him 75 cents for his work. He paid 11 cents per square yard to have it painted and varnished, and it cost him 25 cents to have it brought to his school-room and put up. What was the whole cost of the board?

11. If a piece of board 14 feet long and 13 feet wide costs 40 cents, how much would boards enough to cover a floor 20 feet long and 16 feet wide cost?

12. Isabella had a rectangular flower-bed of which she wished to find the contents. She therefore took a rule 1 feet long, and, by measurement, found that the side of the flower-bed was 8, and the end 3 times as long as the rule. How many square feet did her flower-bed contain?

13. Hannah had a rectangular flower-bed which she measured with a stick 14 feet long. She found the side to be 6, and the end 4 times as long as the stick. How many square feet did her flower-bed contain?

14. Martha had a flower-bed which she measured with a string 25 feet long. She found its side to be, and its end to be as long as the string. How many square feet did her flower-bed contain?

15. Myra had a shawl which was 7 feet long and 64 feet wide, and Ada had one 7 feet long and 6 feet wide. Which contained the most square feet, and how many the most?

16. How many square feet would there be in the outer surface of a box 7 feet long, 5 feet wide, and 4 feet high? How much would it cost to paint it at 12 cents per foot?

Suggestions. The top and bottom of the box will each be 7 feet long and 5 feet wide; the two sides will each be 7 feet long and 4 feet high; and the two ends will each be 5 feet long and 4 feet high. The sum of all these surfaces is the quantity required. Considerations like the following will usually very materially shorten the work in such examples. Since the box is 7 feet long and 5 feet wide, the distance round it must be 7+5+7+ 5 feet 24 feet. The sides and ends of the box, then, will be equivalent to a surface 24 feet long and 4 feet high. The top and bottom, each being 7 feet long and 5 feet wide, will together be equal to a surface 7 feet long and 10 feet wide.

17. How many feet of canvass would be required to cover 3 boxes, each box being 4 feet long, 24 feet wide, and 2 feet high? What would be the cost of the canvass at 2 cents per square foot?

18. A man bought a yard 12 feet long and 8 feet wide for 7 cents per square foot, and built around it a tight board fence 4 feet high. What did the land cost? What did the boards for the fence cost at 1 cents per foot?

19. How many square feet in the walls of an enclosure 29 feet long and 12 feet wide, the walls being 9 feet high, and what would it cost to paint them at 82 cents per square yard?

20. If a lot of land is 8 rods in length, what must be its width that it many contain 1 rood?

Solution. If it were 8 rods long and 1 rod wide, it would contain 8 square rods. Therefore, to contain 1 rood or 40 square rods, it must be as many rods wide as there are times 8 in 40, which are 5 times. Therefore, it must be 5 rods wide. 21. If a rectangular township containing 56 square miles is 8 miles in length, what must be its width?

22. Mr. Shepard owns a square garden containing 81 square rods. What is the length of its side?

23. A certain township in the form of a square contains 36 square miles. What is the length of its side?

24. Jason had a rectangular piece of gold leaf containing 64 square inches, and on measuring it he found that its length and breadth were equal. What was its length?

25. My sitting-room and parlor are each 5 yards wide, but my parlor is 2 yards longer than my sitting-room. The floor of my sitting-room contains 30 square yards. What is the length of my parlor? How many square yards are there in my parlor floor?

26. My house-lot is 12 rods long, but if it were only 8 rods long, it would contain 40 rods less than it now does. How much does it contain ?

27. Mr. Wiswall's house-lot is 11 rods long and 9 rods wide. Mr. Messenger's contains 21 rods more than Mr. Wiswall's, and is 12 rods long. How wide is it?

28. Mr. Hitch's garden is 16 rods long, but if it were 7 rods longer it would contain 63 more square rods. How wide is it? How many square rods does it contain?

29. Mr. Wade owns a square garden, and he finds that if he doubles its length without altering its width, it will contain 72 square rods. How long is the garden?

30. Mr. Jenks says that if his black-board were 2 ft. wider than it now is, it would contain 26 more square feet, but if it were 2

ft. longer it would contain 11 more square feet. What is its length and width, and how many square feet does it contain?

31. How many cubic feet in a block 9 ft. long, 8 ft. wide, and 7 ft. high?

Solution. Since the block is 9 ft. long and 8 ft. wide, there must be 9 times 8, or 72, sq. ft. in the bottom of it. But each foot in height will give 1 cubic foot for every square foot in the bottom; and as in this case there are 72 sq. ft. in the bottom, there must be 72 solid feet for each foot in height. Therefore, 7 ft. in height must give 7 times 72 cubic feet, or 504 cubic ft. Therefore, a block 9 ft. long, 8 ft. wide, and 7 ft. thick contains 504 cu. ft.

32. How many cubic inches in a brick 9 inches long, 4 inches wide, and 2 inches thick?

33. How many cubic feet in a box 16 ft. long, 4 ft. wide, and 3 ft. thick?

34. How many cubic feet in a pile of wood 32 ft. long, 4 ft. wide, and 3 ft. high? How many cord feet? How many cords? 35. How many cubic feet in a stick of timber 15 ft. long, 2 ft. wide, and 13 ft. thick?

36. Mr. Wales was offered his choice of 2 piles of wood, one of which was 14 ft. long, 8 ft. wide, and 4 ft. high, and the other was 16 ft. long, 6 ft. wide, and 4 ft. high. Not knowing how to calculate, he chose the smaller pile. Which pile did he choose, and how many cubic feet of wood did he lose by his ignorance? How many cord feet? If the wood was worth $4 per cord, how many dollars' worth did he lose?

37. How many more cubic feet will a bin 18 ft. long, 8 ft. wide, and 6 ft. high contain, than one 21 ft. long, 9 ft. wide, and 3 ft. high?

38. If a block of wood 3 ft. long, 2 ft. wide, and 13 ft. thick is worth 72 cents, how many cents is a block 9 ft. long, 24 ft. wide, and 12 ft. thick worth?

39. Mr. Bowen has a pile of wood 9 ft. long, 4 ft. wide, and 5 ft. high. Mr. Jackson has a pile the width and height of which are the same with Mr. Bowen's, but the length is twice as great. How many times as many cubic feet are there in Mr. Jackson's pile as in Mr. Bowen's? How many cubic feet in each?

40. Mr. Gordon has a pile of wood whose height and length are the same as the height and length of Mr. Jackson's, but whose width is twice as great. How many times as many cubic feet does it contain as Mr. Jackson's? as Mr. Bowen's? How many cubic feet does it contain?