4. How many square feet in a gate 12 feet 5 inches wide, and 6 feet 8 inches high? 5. How many square feet in a floor 16 feet 6 inches long, and 12 feet 9 inches wide ? 6. How many square feet in a ceiling 53 feet 6 inches long, and 25 feet 6 inches wide ? 7. How many square feet are there in a stock of 6 boards 17 feet 7 inches long, and 1 foot 5 inches wide ? 8. How many feet in a stock of 10 boards 12 feet 8 inches long, and 1 foot 1 inch wide ? 9. How many cubic feet in a stick of timber 12 feet 10 inches long, 1 foot 7 inches wide, and 1 foot 9 inches thick? * 10. How many cubic feet in a block of marble 8 feet 4 inches long, 2 feet 6 inches wide, and 1 foot 10 inches thick? 11. How many cubic feet in a load of wood 6 feet 7 inches long, 3 feet 5 inches high, and 3 feet 8 inches wide ? 12. How many feet in a load of wood 7 feet 2 inches long, 4 feet high, and 3 feet wide ? 13. How many feet in a load of wood 9 feet long, 4 feet 3 inches wide, and 5 feet 6 inches high? 14. How many feet in a pile of wood 100 feet long, 5$ feet high, and 4 feet wide ? 15. How many feet in a pile of wood 150 feet long, 8£ feet high, and 5 feet wide ? 16. How many cubic feet in a wall 40 feet 6 inches long, 5 feet 10 inches high, and 2 feet thick ? 17. How many solid feet in a vat 10 feet 8 inches long, 7 feet 2 inches wide, and 6 feet 4 inches deep? 18. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, are there in a wall 20 feet long, 10 feet high, and 14 feet thick ? 19. How much will the flooring of a room which is 20 feet long and 18 feet wide come to, at 6$ cents per square foot ? 20. How much will the plastering of a room 16 feet square come to, at 12 cents per square yard ? SECTION XIV. POWERS AND ROOTS. INVOLUTION. Art. 337. Ex. 1. What is the product of 5 multiplied by 5 ? Ans. 5X5=25. 2. What is the product of 3 multiplied into 3 twice? Ans. 3X3X3=27. 3. What is the product of 2 into 2 three times ? Ans. 2x2x2x2=16. 338. When any number or quantity is multiplied into itself, the product is called a power. Thus in the examples above the products 25, 27, and 16 are powers. 339. Powers are divided into different orders ; as the first, second, third, fourth, fifth power, &c. They take their name from the number of times the given quantity or number is used as a factor, in producing the given power. The original number, that is the number which being multiplied into itself, produces a power, is called the first power or root of all the other powers of that number; because they are derived from it. 3 yards. Note.-1. The second power of a number is also called the square ; (Art. 153. Obs. 1 ;) for if the side of a square is 3 yards, then the product of 3X3=9 yards, will be the area of the given square. (Art. 163.) But 3X3 9 is also the second power of 3 ; hence it is called the square. 3X3=9 yards. 3 yards. QUEST.--338. What is power? 339. How are powers divided? From what do they take their name? What is the first power? Note. What is the second power called ? The third ? The fourth? 2 feet. 2. The third power of a number is also called the cube ; (Art. 154. Obs. 2 ;) for if the side of a cube is 2 feet, then the product of 2X2X2= 8 feet, will be the solidity of the given cube. (Art. 164.) But 2X2X2=8, is also the third power of 2 ; hence it is called the cube. 3. The fourth power of a number is called the biquadrate. 2 feet. 2x2x2=8 feet. MENTAL EXERCISES. 4. What is the square of 4 ? Ans. 16. 5. What is the cube of 3 ? The fourth power of 3 ? 6. The fourth power of 2? The fifth power of 2 ? 7. What is the square of 5? Of 6 ? Of 7 ? Of 9 ? Of 8 ? Of 10 ? Of 11 ? Of 12 ? 8. What is the cube of 3? Of 4? Of 5 ? Of 6 ? 340. The number of times a quantity or number is employed as a factor to produce the given power, is usually denoted by a small figure placed above it at the right hand. This figure is called the index or exponent. The index of the first power is 1 ; but this is omitted ; for (2)=2. The square or second power of 4, or 4 x 4, is written (4)?; the cube or third power of 12, or 12 x 12 x 12, is written 12). Hence, 21=2, the first power of 2; 22=2x2, the square, or 2d power of 2; 23=2x2x2, the cube, or 3d power of 2; 24=2x2x2x2, the biquadrate or 4th power of 2; 25 =2x2x2x2x2, the 5th power of 2. Quest.-340. How are powers denoted? What is this figure called What does it show? What is the index of the first power? EXERCISES FOR THE SLATE. 9. Express the third power of 6; the 4th power of 12. 10. Express the square of 16; the cube of 20; the fourth power of 25; the fifth power of 72 ; the sixth power of 100; the tenth power of 500. 341. The process of finding a power of a given number by multiplying it into itself, is called Involution. 342. Hence, to involve a number to any required power. Multiply the given number into itself, till it is taken as a factor, as many times as there are units in the index of the power to which the number is to be raised. (Art. 339.) Obs. 1. The number of multiplications in raising a number to any given power is one less than the index. Thus the square of 3 is written 3?, and 3X3=9, the 3 is taken twice as a factor, but there is but one multiplication. 2. A Fraction is raised to a power by multiplying it into itself. Thus the square off is :X}= Mixed numbers may be reduced to improper fractions, or the common fraction may be reduced to a decimal. 3. All powers of 1 are the same, viz: 1; for 1x1x1x1, &c.=1. 11. What is the square of 24 ? Common Operation. Analytic Operation. 24 24=2 tens or 20+4 units. 24 24=2 tens or 20+4 units. 96 80+16 48 400+80 576 Ans. And 400+160+16=576. It will be seen from this operation that the square of 20+4, contains the square of the first part, viz: 20 x 20 =400, added to twice the product of the two parts, viz: 20 X 4+20 X4=160 added to the square of the last part, viz : 4X4=16. Hence, QUEST.-341. What is involution ? 342. How is a number involved to any required power ? Obs. How many multiplications are there in raising a number to a given power? How is a fraction involved? A mixed number? What are all powers of 1? 342. a. The square of any number which consists of two figures is equal to the square of the tens, added to twice the product of the tens into the units, added to the square of the units. Note.-1. The product of any two factors cannot have more figures than both factors, nor but one less than both. For example, take 9, the greatest number which can be expressed by one figure. (Art. 7.) And (9)2, or 9X9==81, has two figures, the same number which both factors have. 99 is the greatest number which can be expressed by two figures ; (Art. 7 ;) and (99)2, or 99 X99=9801, has four figures, the same as both factors have. Again, 1 is the smallest number expressed by one figure, and (1)2, or 1X1=1, has but one figure less than both factors. 10 is the smallest number which can be expressed by two figures; and (10)2, or 10x10=100, has one figure less than both factors. Hence, 2. Any square number cannot have more figures than double the number of the root or first power, nor but one less. 3. A cube cannot have more figures than triple the number of the root or first power, nor but two less. 12. What is the square of 45 ? 50 ? 75 ? 100 ? 540 ? 13. What is the cube of 5 ? Of 8? 10 ? 12? 60 ? 14. What is the fourth power of 3? Of 4 ? 16 ? 20 ? 15. What is the fifth power of 2 ? Of 3 ? 4? 5? 6? 16. What is the square of { ? Of {? £? ? $? ? 17. What is the cube of ß? Of ? of 7? Of ? 18. What is the square of 2}? Of 37 ? 5 ? 104? 19. What is the square of 1.5 ? Of 3.25? Of 10.25 ? EVOLUTION. 343. If we resolve 25 into two equal factors, viz: 5 and 5, each of these equal factors is called a root of 25. So if we resolve 27 into three equal factors, viz: 3, 3, and 3, each factor is called a root of 27; if we resolve 16 into four equal factors, viz: 2, 2, 2, and 2, each factor is called a root of 16. And, universally, when a number is resolved into any number of equal factors, each of those factors is said to be a root of that number. Hence, Quest.--342. a. What is the square of any number consisting of two fig. ures equal to ? Note. How many figures are there in the product of any two factors? How many figures will the square of a number contain ? The cube? 343. When a number is resolved into any number of equal factors, what is each of those factors called ? |