7. Required the 4th root of a-8ab+24al'-32ab+166*. 8. Required the 6th root of a1+6a1b+15a°b3+20a*b*+15a*b* +6a2b*+b°. Ans. a-2b. Ans. a+b. GENERAL THEORY OF EXPONENTS. 261. It has already been shown that am an am Xan = am+n, =a", and (a")"=aTM", m and n being integers, and either positive or negative. To prove that the above relations are true universally, it remains only to show that they hold true when m and n are fractional. Reducing the exponents to a common denominator, we have But from the nature of fractional exponents, (222), the second member of this equation may be written and as the two factors have the same radical index (227) the result reduces to II. To show that a'÷a' = By transformations similar to those just employed, we have hence, by equating values of x in (1) and (5), (4) (5) We conclude, therefore, that in multiplication, division, involution and evolution, the same rule will apply, whether the exponents are positive or negative, integral or fractional EXAMPLES. 1. Multiply a by a3, and simplify the product. X = 7 (a3×a3) × (b3×b3) = a3b = alva, Ans. 2. Simplify the expression (x3×x3)2. (x3×x3)3 = (x¦§)3 = x3, Ans. 3. Multiply x—3x2+xa by x1—2x1—3. OPERATION. x_2x_3 -2x+6x-2x -3x2+9x3-3x2 xa—5x+4x3+7xa—3x1, Ans. 4. Divide x-5√x*—5¥x3—5√/x—6x by ✅x3-2√/x+ 8x. 7. Find the product of a, a, a, and a. 8. Divide ac3 by a3ç3 Ans. a'Vab. Ans. Va. 3 Ans. aa—3+3a ̄†—at. Ans. at-at 15. Multiply a3+a2b3+a3¿3⁄4+aï3+aa¿§+b2oa by a -3 16. Divide x1+x3a+a3 by x33+x3a3+aa. Ans. a'-l'. Ans. x3—x3a3+a3 ̧ Ans a Ans. 2±√/5. Ans. (2) Ans. V. Ans. ¡(2†5—5√2)‡. (1/5+2)(*/5+1/2)(V5—1/2) 23. Simplify {(13+3) (13+1/3) (√13—1/8) Ans. . 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 a', 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, (-a)' = +a3. Such Hence the indicated root, Va2, is not real but imaginary. 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 Vax(−1) = √a⋅√=1; α also, V -a'—b2+2ab = √(a—b)3× (−1) = (a—b)V—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, V=1, 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 1, and deduce the law of their formation. |