Thus, to raise any power of a quantity to any other .power, its original index must be multiplied by the index of the power to which it is to be raised. 199. To find (ab)". (ab)m = ab × ab × ab... to m factors, by definition, by definition. =(aaa... to m factors) x (bbb... to m factors) [Art. 52.] (abc)" = abc abc × abc... to m factors, Hence = = (aaa..... to m factors) × (bbb... to m factors) =am xbm x cm. X (ccc... to m factors) (abe)" = ambmcm, and so on, however many factors there may be in the expression whose mth power is required. Thus, the mth power of a product is the product of the mth powers of its factors. 200. The most general monomial expression is of the form abc... Thus, any power of a monomial expression is obtained by taking each of its factors to a power whose index is the product of its original index and the index of the power to which the whole expression is to be raised. 202. It should be noticed that all powers of a positive quantity are positive, and that successive powers of a negative quantity are alternately positive and negative. This follows at once from the Law of Signs; for we have (a)2 = (-a) (a) = + a2; and so on. Thus (− a) 3 = ( − a)2 ( − a) = ( + a2) ( − a) — — a3 ; (− a) * = (— a) 3 ( − a) = ( — a3) ( − a) = + a1; (-a) 2n+a2n, and (-a) 2n+1 = — a2n+1. = From the above it is clear that all even powers, whether of positive or of negative quantities, are positive, and that all odd powers of any quantity have the same sign as the original quantity. 203. We have already proved the following cases of the involution of binomial expressions. and (a + b)2= a2+2ab+b2, (a + b)3 = a3 +3a2b+3ab2 + b3. If we multiply again by a + b, we shall have (a + b) = a*+4a3b+6a2b2+4 ab3 + b*. By multiplying the last result by a + b we should obtain (a+b), and by continuing the process we could obtain any required power of a+b; but to find in this way any high power, for instance to find (a + b)20, would clearly be very laborious. We shall shortly prove a theorem, called the Binomial Theorem, which will enable us to write down at once any power of a binomial expression. The above formulæ are identities, and are true for all values of a and b; hence we can write down the squares and the cubes of any binomial expressions. 204. An important case of involution is considered in Art. 70, where the square of a multimonial expression is obtained. EXAMPLES LXI. Write down the value of each of the following: 21. (a+b3)3. 22. (2 a23b2)8. 33. (x3 + x2 - 2 x − 2)2. 34. (a + 2b+3c + 4 d)2. 35. (2 a b + c − 2 d)2. 36. (x2+x+1)3. 37. (x2 -x + 2)3. 38. (3x2-5x+1)3. 2 17. (a5-2b+)2. 2 x THE BINOMIAL THEOREM. 205. The numerical coefficients on the right side of the formulæ of Art. 203 may be so constructed as to exhibit the law of their formation. Thus, written in the order of their occurrence in the formulæ, they and their reconstructed equivalents are If this law of their formation obtains for higher powers of (a+b), we shall be able to write out, by a very simple rule, the entire series of terms in the expansion of (a+b)", wherein n is any positive integer. Thus, if the law holds for all positive integral values of n, which is called the general term, may be made to assume in succession the form and value of every term in the series, by giving to r the successive values. 0, 1, 2, 3, 4, ... n-1, n. |