EXAMPLES. 1. Multiply ab3 by a3b3, and simplify the product. a‡b3×a3¿3 = (a3×a3) × (b3×b3) = a3b = ab1⁄4a, Ans. 2. Simplify the expression (x3×x3)‡. (x3×x})} = (x1§)* = x3, Ans. 3. Multiply xa—3x2+xa by xa—2x1—3. x-3x+x -2x+6x3-2x -3x2+9x-3x+ xa¤—5x+4xa+7xa—3x*, Ans. 4. Divide x-5/x*—5ïx3—5√/x—67x by x-2y/x+ 5. Multiply x by x3. 6. Multiply a2¿ by ažbi. 7. Find the product of aa, a‡, a3, and a→. 8. Divide ac3 by act 12. Multiply a—2a+a1 by aa—a. 13. Divide a-b by va+vb. 14. Divide a 2a+aš by a3 —1. Ans. a3—3+8a¤—a†. 8 Ans. Va―√b. Ans. at—að 15. Multiply a2+ab+a3b3+ab3+ab+b23° by aa—¿§. 16. Divide x+x3a+a3 by x2+x3a3+a3. Ans. a3-b'. 22. Simplify (Va+v/b) (va—√b) Ans. √a+vb. { (√/5+2)(V5+1/2)(√5—√/2) 28. Simplify (18+) (13+3) (√18-7/8) IMAGINARY QUANTITIES. 262. It has been shown (228, 3) that an even root of a negative quantity is imaginary, an expression for such a root being a symbol of an impossible operation. Thus if we take a2, which is numerically a perfect square, and affect it with the minus sign, we can not obtain the square soot of the result. For we have (+a)2 = +a3, Hence the indicated root, V-a, is not real but imaginary. Such expressions are, however, of frequent occurrence in analysis and its application to physical science, and conclusions of the highest importance depend upon their use and proper interpretation. We therefore proceed to investigate the rules to be observed in operating with such quantities. 263. When a real and an imaginary quantity are connected in a single expression, the whole is considered imaginary on account of the presence of the imaginary part. Thus the binomial, 4+1-3, considered as a single quantity, is imaginary. 264. According to (227), we may have also, √—a = √ a×(−1) = √a · √=1 ; √—a2—b2+2ab = √ (a—b)*X(−1) = (a—b)√—1. Hence, if we regard only quadratic expressions, every imaginary quantity may be reduced to the form, a±bv=1, in which a is the real part, b the coefficient of the imaginary part, and V1 the imaginary factor. Thus we may employ only the single symbol, V1, to indicate that a quantity is imaginary. 265. For convenience in multiplication and division of imaginary quantities, we will now obtain some of the successive powers of the symbol V1, and deduce the law of their formation. Multiplying these powers, in their order, by the 4th, we shal obtain the 5th, 6th, 7th and 8th, the same as the 1st, 2d, 3d, and 4th; and so on. 266. The common rules for multiplication and division of radicals will apply to imaginary quantities, with a simple modification respecting the law of signs. Let it be required to find the product of Va and Vb. To obtain the true result, we must separate the imaginary symbol V1 from each factor. Thus But if we multiply by the common rule for radicals, (253), we shall have v=axv=b = √(———-a) · (-—-—-—b) =√ ab, a result erroneous with respect to the sign before the radical. Proceeding as in the first operation we find that (√=a)X(-b) = +√ ab ·⋅ (−1) = −√ ab ; Thus, like signs produce, and unlike signs produce +. Hence, 1.- The product of two imaginary terms will be real, and the sign before the radical will be determined by the common rule reversed. We may operate in like manner in division of imaginary quantities. Thus, That is, like signs produce + and unlike signs produce. Hence, 2. The quotient of one imaginary term divided by another will be real, and the sign before the radical will be governed by the common rule. 267. Let us assume the equation (2) in which a and a' are real. By transposition, a-a' = · (b'—b)√—1. Now it is evident that in this equation a = a'. For, if a> a' or a <a', then the first member of equation (2) is different from zero, and real. But this can not be, because the second member is either nothing or imaginary. Hence a a'; and equation (2) becomes 0 = (b'—b)√—1, which can only be satisfied by putting b = b'. Hence, If two imaginary quantities are equal, then the real parts are equal, and the coefficients of the imaginary symbol are also equal. 268. These principles may now be applied in the following 6. Multiply a +V=c by Va+√/c. Ans. (a+c)√=1. C 7. Divide 91—10 by 3√—2. 8. Divide av by cV―d. Ans. 3/5. a 16 Ans. с |