141. By using brackets the involution of a quantity consisting of more than two terms may be always made to depend upon that of a binomial, and thereby the operation be much abridged. 142. Since (a+b+c+d+&c.) may be thus arranged, a2+2a(b+c+d+&c.)+b2+2b(c+d+&c.)+c2+2c(d+&c.)+&c. the square of any multinomial may be readily found by the following RULE: Square each term, and multiply twice that term into the sum of the several terms that come after; the sum of all the results so obtained will be the square of the whole quantity. Thus, 11 1 Ex. 4. (1+x+2x2+ 3x3+ 4x*+5x3+.....)2 =1+2(x+2x2+3x3+ 4x*+5x3+6x®+.....) +x2+2x(2x2+3x3+4x*+5x3+.....) +9x+6x3(4x*+...) +...... .... =1+2x+5x2+10x3+18x1+30x3+47x®+... N. B. It will be found useful to commit to memory the following results: (A+B)3=A3+B3+3AB(A+B)................(1), whatever quantities A and B may represent, simple or compound. Thus, Ex. 1. Let the cube of 1+x+x2 be required. (1+x+x2)3= (1+x+x2)3, =(1+x)3+ (x2)3+3(1+x)x2(1+x+x3), by (1), =1+x2+3x+3x2+x®+3x2+3x3+3x2+3x3+3x2+3x3, =1+3x+6x2+7x3+6x1+3x3+x®. Ex. 2. Required the cube of Ja+x-Ja-x. (√a+x−√a−x)3=a+x-a-x-3√a+x.√a-xx(Ja+x-Ja-x), by (2), =2x−3.√a+x.√a−x.{√a+x−√a−x}. 143. EVOLUTION, or the extraction of roots, is the method of determining a quantity which raised to a proposed power will produce a given quantity. 144. Since the nth power of a" is am" (Art. 138), the nth root of aTM must be am; that is, to extract any root of a simple quantity, we must divide the index of that quantity by the index of the root required. Thus Jaa', Ja1a=a3; &c. "√aTM"=aTM. 145. When the index of the quantity is not exactly divisible by the number which expresses the root to be extracted, that root must be represented according to the notation pointed out in Art. 70. Thus the square, cube, fourth, nth, root of a2+x2, are respectively represented by (a2+x2)1, (a2+x')‡, (a2+x2)‡, (a2+x2)2; the same roots of 1 a2 + x2 or (a2 +x2)−1, are represented by (a2 + x2) −1, (a2 + x2) −1, (a2 + x2)−1‚ (a2+x2) ̃ ̄*. 146. If the root to be extracted be expressed by an odd number, the sign of the root will be the same with the sign of the proposed quantity, as appears by Art. 137. Thus 8 is 2; -8 is -2; &c. 147. If the root to be extracted be expressed by an even number, and the quantity proposed be positive, the root may be either positive or negative. Because either a positive or negative quantity raised to such a power is positive (Art. 137). Thus a is a; √(a+x) is (a+x)3; &c. 148. If the root proposed to be extracted be expressed by an even number, and the sign of the proposed quantity be negative, the root cannot be extracted; because no quantity raised to an even power can produce a negative result. Such roots are called impossible. 149. Any root of a product may be found by taking that root of each factor, and multiplying the roots, so taken, together. 1 11 Thus (ab)" = a" ×b"; because each of these quantities, raised to the nth power, is ab (Art. 138). Exs. Jab Jabab'; and ab13c3=a2b3c. COR. r+s a" × a"=a". 1 1 2 If a=b, then a"xa"a"; and in the same manner This will be proved more fully and clearly in Art. 162. 150. Any root of a fraction may be found by taking that root of both the numerator and denominator (Art. 139), that is, the root of the numerator for a new numerator, and the same root of the denominator for a new denominator. 151. To extract the square root of a compound quantity. Since the square root of a2+2ab+b2 is a + b (Art. 140), whatever be the values of a and b, we may obtain a general rule for the extraction of the square root, by observing in what manner a and b may be derived from a'+2ab+b2. a2+2ab+b2 (a+b a2 2a+b)2ab+b2 2ab+b2 Having arranged the terms according to the dimensions of one letter, a, the square root of the first term, a2, is a, the first term in the root; subtract its square from the whole quantity, and bring down the remainder 2ab+b2; divide 2ab by 2a, and the result is b, the other term in the root; then multiply the sum of twice the first term and the second, 2a+b, by the second, b, and subtract this product, 2ab+b2, from the remainder. If there be more terms, consider a+b as a new value of a; and its square, that is, a2+2ab+b2, having by the first part of the process been subtracted from the proposed quantity, divide the remainder by the double of this new value of a, for a new term in the root; and for a new subtrahend multiply this term by twice the sum of the former terms increased by this term. The process must be repeated till the root, or the necessary approximation to the root, is obtained *. Ex. 1. 2bc + c2. To extract the square root of a2+2ab+b2+2ac+ Having arranged the terms of the proposed quantity according to the dimensions of one letter, a, it becomes a2+2(b+c)a+b2+2bc + c2. • That the Rule may be thus extended will be obvious from comparing the form of the squares of a+b+c, a+b+c+d, &c. with that of a+b, from which the Rule was deduced. For (a+b+c)2=(a+b)2+2(a+b)c+c2, (Art. 141) =a2+(2a+b)b+{2(a+b)+c} c. (a+b+c+d)2=(a+b+c)3+2(a+b+c)d+d3, =a2+(2a+b) b+{2(a+b)+c} c+{2(a+b+c)+d}d; and so on; from which method of exhibiting the square of a multinomial the rule for extracting the square root is evidently seen to hold whatever be the number of terms in the root.-ED. |