(45.) To Multiply fractional quantities. Rule. Multiply the numerators together for a new numerator, and also the denominators for a new denominator; and reduce the products to their lowest term. (46) To divide fractional quantities. Rule. Invert the divisor, and proceed, in all respects, as in Mul tiplication. (Art. 45.) (47.) SCHOLIA 1. If the fractions to be divided have a common denominator take the numerator of the dividend for a new numerator, and the numerator 3 of the divisor for the denominator; for it is evident, that are contained 12 9 12 as many times in as there are 3's in nine; i. e. thrice. 2. When a fraction is to be divided by any quantity, it is the same thing whether the numerator be divided by it, or the denominator multiplied by it. α For, if it be proposed to divide by c, we change it into a viding the ac by c, we have for the quotient sought. a be fraction is to be divided by c, we have only to that quantity, and leave the numerator as it is. Thus 16-5-- 80° ac and then di bc 3. When the two numerators or the two denominators are divisible by a common quantity, we expunge that quantity from each, and use the quotients in place of the fractions first proposed. 4. The signs + and cation; for 2 preserve the same laws in division as in multipli 15 3 3 12 19 INVOLUTION. (18.) Involution, or the raising of quantities to a given power, is performed by the continued multiplication of a quantity into itself, till the number of factors amounts to the number of units in the index of the given power; or it is the method of finding the square, cube, biquadrate, &c. of any given quantity or root. Obs. These operations may be easily performed upon small numbers and simple algebraic quantities, as will appear from the mere inspection of the following tables, which also illustrate the definition we have given of this branch of algebraic computation. 6561 10000 4th Power 1 16 81256 625 1296| 2401 4096 5th Power 132 243 102431257776 16807 | 32768 | 59049 100000 Illus.—If I want to find the fifth power of the number 6, I proceed thus Illus. If it is required to find the 5th power of — b, then -b== the root. b = bx bx -E the cube. = W H 36 is expanded thus: 2a 32a 36 = 24365 the 5th power. 36 H 24 36 2a 36 X 36 2a 36 2a 36 Note. When the root is +, all the powers of it will be positive, and when it is negative the odd powers will take the sign, and the even powers the sign+; and from these tables we derive the following rule for finding the power of any quantity. (49.) Rule. Multiply, or involve, the quantity into itself to as many factors as there are units in the index, and the last product will be the power required. Or, Multiply the index of the quantity by the index of the power, and the result will be the same as before. (50.) SCHOLIUM. In the involution of a binomial quantity of the form a + b, the component terms of the successive powers will be found to bear a certain relation to each other, and to observe a certain law, according to the incex of the given power. To illustrate this, we inspect the following table: 6th Power. + a +6ab+15ab2+20ab+15a2ba +бab® +b® Obs. The successive powers of a- b are the same as those of +b, except that the sign of the terms will be alternately +and lus. In examining that column of the foregoing table which is occupied by the expanded powers of a + b, we discover, 1. That in each case, the first term is a raised to the given power; and the last term is b raised to the given power. For in the square, a2 is the first, and 6 is the last terna: in the cube, a is the first, and the last, &c. 2. That in the intermediate terms, the powers of a decrease, while the powers of b increase, by unity, in each successive term. 3. That in each case, the co-efficient of the second term is the same with the index of the given power. 4. That if the co-efficient of a in any term be x by its index, and the product by the number of terms to that place, the quotient will give the co-efficient of the next term. For example, in the 4th power, the co-efficient of a in the 2d term x its index 4 12 number of terms to that place 6 co-efficient of the 3d term. Where 4 is the co-efficient of a, 3 is its inder, and 2 denotes the number of terms. |