If a floor be 10 ft. 4' 5" long, and 7 ft. 8' 6" wide, what is its surface ? Ans. 79 ft. 11' 0" 6'' 6"". What is the solidity of a wall 53 ft. 6' long, 10 ft. 3' high, and 2 ft. thick ? Ans. 10964 ft. INVOLUTION. When a number is multiplied into itself, it is said to be involved, and the process is called Involution. Thus 2x2x2 is 8. Here the number 2 is multiplied into itself twice. The product which is obtained by multiplying a number into itself, is called a Power. Thus, when 2 is multiplied into itself once, it is 4, and this is called the second power of 2. If it is multiplied into itself twice (2X2X2=8 the answer is 8, and this is called the third power. The number which is involved, is called the Root, or first power. Thus, 2 is the root of its second power 4, and the root of its third power 8. A power is named, or numbered, according to the number of times its root is used as a factor. Thus the number 4 is called the second power of its root 2, because the root is twice used as a factor; thus, 2 X2=4. The number 8 is called the third power of its root 2; because the root is used three times as a factor ; thus, 2x2x2=8. The method of expressing a power, is by writing its root, and then above it placing a small figure, to show the number of times that the root is used as a factor. Thus the second power of 2 is 4, but instead of writing the product 4, we write it thus, 22. The third power of 2 is written thưs, 23. The small figure that indicates the number of times that the root is used as a factor, is called the Index, or Expo. nent, What is involution? What is a power ? root ? How is a power named ? The different powers have other names beside their numbers. Thus, the second power is called the Square. Powers are indicated by exponents. When a power is actually found by multiplication, involution is said to be performed, and the number or root is involved. RULE OF INVOLUTION. To involve a number, multiply it into itself, as often as there are units in the exponent, save once. NotE.—The reason why it is multiplied once less than there are units in the exponent, is, that the first time the number is multiplied, the root is used twice as a factor ; and the exponent shows, not how many times we are to multiply, but how many times the root is used as a factor. 1. What is the cube of 5 ? Ans. 5x5x5=125. 2. What is the fourth power of 4 ? Ans. 256. 3. What is the square of 14 ? Ans. 196. 4. What is the cube of 6 ? Ans. 216. 5. What is the 5th power of 2 ? Ans. 32. 6. What is the 7th power of 2 ? Ans. 128. 7. What is the square of } } Ans. .. 8. What is the cube of f? A Fraction is involved, by involving both numerator and denominator. 9. What is the fourth power of z? Ans. 1 10. What is the square of 54 ? Ans. 304. 11. What is the square of 304 ? Ans. 91516 12. Perform the involution of 85. Ans. 32,768. 13. Involve zy, lj, and to the third power each. Ans. 3o; Hoti 4. 14. Involve 2113. Ans. 9,393,931. 15. Raise 25 to the fourth power. Ans. 390,625. 16. Find the sixth power of 1.2. Ans. 2.985,984. What are the names of the different powers ? What is the rule for involution ? How is a fraction involved? Ans. 625 EVOLUTION. Evolution is the process of finding the root of any num. ber ; that is, of finding that number which, multiplied into itself, will produce the given number. The Square Root, or Second Root, is a number which be. ing squared (i. e. multiplied once into itself) will produce the given number. It is expressed either by this sign, put before a number, thus V4, or by the fraction placed above a number, thus, 41. The Cube Root, or Third Root, is a number, which be. ing cubed, or multiplied by itself twice, will produce the given number. It is expressed thus, Ņ 12; or thus, 12). All the other roots are expressed in the same manner. Thus the fourth root has this sign put before a number, or else 4 placed above it. The sixth root has before it, or ! above it, &c. There are some numbers whose roots cannot be pre. cisely obtained; but by means of decimals, we can approximate to the number which is the root. Numbers whose roots can be exactly obtained, are called rational numbers. Numbers whose precise roots cannot be obtained, are called surd numbers. When the root of several numbers united by the sign + or is indicated, a vinculum, or line is drawn from the sign of the root over the numbers. Thus, the square root of 36—8 is written 36—8. The root of a rational number, is a rational root, and the root of a surd number, is a surd root. It is very necessary for practical purposes, to be able to find the amount of surface there is in any given quantity. For instance, if a man has 250 yards of matting, which is 2 yards wide, how much surface will it cover ? The rule for finding the amount of surface, is to multiply the length by the breadth, and this will give the amount of square inches, feet, or yards. What is evolution? How are roots expressed ? What are rational numbers ? What are surd numbers? What is a rational root? Surd root? What is the rule for finding the amount of surface ? It is important for the pupil to learn the distinction be. tween a square quantity, and a certain extent that is in the form of a square. For example, four square inches, and four inches square are different quantities. A Four square inches may be represented in Fig. A. In this figure there are four square inches, but it makes a square which is only two inches on each side, or a two inch square. B. A four inch square may be re. presented by Fig. B. Here the sides of the square are four inches long, and it is called a four inch square. But it con. tains sixteen square inches. For when the four inch square is cut into pieces of each an inch square, it will make sixteen of them. A four inch square then, is a square whose sides are four inches long. Four square inches are four squares that are each an inch on every side. When we wish to find the square contents of any quan. tity, we seek to know how many square inches, or feet, or yards, there are in the quantity given, and this is always found by multiplying the length by the breadth. When the length and breadth of any quantity are given, we find its square contents, or the amount of surface it will cover, by multiplying the length by the breadth. What are the square contents of 223 yds. of carpeting wide ? What are the sq. contents of 249 yds. of matting & wide ? If any quantity is placed in a square form, the length of one side is the square root of the square contents of this figure. Thus in the preceding example, B, the square contents of the figure are 16 square inches. The side of the square is 4 inches long; and 4 is the square root of 16. What is the difference between an inch square, and a square inch ? The square root, therefore, is the length of the sides of a square, made by the given quantity. If we have one side of a square given, by the process of Involution, we find what are the square contents of the quan. tity given. If, on the contrary, we have the square contents given, by the process of Evolution, we find what is the length of one side of the square, which can be made by the quantity given. Thus if we have a square whose side is four inches, by Involution we find the surface, or square contents to be 16 square inches. But if we have 16 square inches given, by Evolution we find what is the length of one side of the square made by these 16 inches. EXTRACTION OF THE SQUARE ROOT. Extracting the square root is finding a number, which, multiplied into itself, will produce the given number; or, it is finding the length of one side of a certain quantity, when that quantity is placed in an exact square. It will be found by trial, that the root always contains just half as many, or one figure more than half as many figures as are in the given quantity. To ascertain, there. fore, the number of figures in the required root, we point off the given number into periods of two figures each, be. ginning at the right, and there will always be as many figures in the root as there are periods. 1. What is one side of a square, containing 784 square feet ? 784(2 Pointing off as above, we find that the root will 4 consist of two figures, a ten and a unit. 384 Repeat the explanation of the rule for extracting the square root. |