Vas, ai m m m m n2 n a X a n ха m m to n terms n an a" m an pro denominator the root which must be taken of the power so obtained. Thus, ał = Va', as = Yao = a*; and generally a" = Mam. The above definition is that which follows at once if we assume the law proved in Art. 24, page 159, viz., am * a" = am +" to be true, whatever be the value of m and n. Thus we have(a*)" ...to n factors + m + = am. Hence, taking the nth root, Nam 10. To show that (at): = at (a}); = (at); by Def., Art. 9. () = (vai), or, raising each side to the qth power, we have (a")" = a PT. Hence, taking the (98)th root, we have x = a n 11. To show that añ x bö = (ab)*. Now, an x b ñ mo Vam / am xmlīm = ambm = "/(ab)" '= (ab)", by Def., Art. 9. And so, a + bä = (*)*. Ex. 1. Multiply together albict by abbtc4. Adding together the indices of like letters, we have * + } = %, } + t = 185, + Ito Hence, the required product is a6b1%cht, Let x = .. 2018 2098 = Ex. 2. Multiply a + ały! + y by x - ały + y. x + xyz + y xtyt + y + XY ху + ya cy! otyš - y Y Ex. V. + y^1. Find the value of_ 1. (ao)}, af, (a-2)-1, (af)#. 2. (a + x)}(a + 2 ax + 20°)}, (at - æž)} (at + æt). Multiply together3. al aix} + xś and ał + atc} + 2* 4. * - ył and x-+ 5. x + ažyt + y and ac + y + + y^? Divide6. a3 – 6 by a 63, at - bi by aš - 6. 7. # - myt + aty ayt + ały - ył by x3 + -ty. 8. a + b + c – 3 a£b$ct by a} + + cs. 9. at - a -4 - atb + by abbt - 6-5, , m m m m m m 12. (2m - am) → (2.* (ac? + a“) (2c* + a 4) (208 + an) (202" + a?”). Find the square roots of 13. a + b + c + 2 (ałb? + ałct + 6că). 14. 4 xyš – 12 zły + 17 x*y$ – 12 x{y} + 4x? 15. a?-1 - 4 ab-1-8a-17% + 4a-% + 8. Find the cube roots of — 16. 204 + 9.Y + 6C5 - 99 x* - 42 25 + 4412ct - 343. 17. Rog-1 + 3 ca-3 + 3 ga-3 + 1. 18. ab (1 + 3 a-36} + 3 a-{b} + a-18) (ab-1 - 3 až6 –} + 3 ałb-} + 1). CHAPTER IV. SURDS. 12. A surd quantity is one in which the root indicated cannot be denoted without the use of a fractional index. Thus, the following quantities are surds : Na ac + Y Yx a 1 Since, from what has been explained in the last chapter, these quantities may be written thusat, (a+ x)}, (a + b 2 + c?)", * (a + x)* (x + y)} it follows that surds may be dealt with exactly as we deal with their equivalent expressions with fractional indices. It is evident that rational quantities may be put in the form of surds, and conversely, expressions which have the form of surds may sometimes be rational quantities. Thus, ao = Va)3 = Jas; and I a3 + 3 ab + 3 ab? + 13 Va + b)3 = a + b. 13. A mixed quantity may be expressed as a surd. Thus, 3 V5 = 33. 35 = 33 x 5 = 135, and so, w my 14. Conversely, a surd may be expressed as a mixed quantity, when the root of any factor can be obtained. Thus, V18 a:h2 = 19 a’62 2 a V9 a®6. J2a = 3 ab 12 a. Na + b*)$z*y3. Væra (a2 + 62)*xy Way. 15. Fractional surd expressions may be 80 expressed that the surd portion may be integral. The process is called rationalizing the denominator, and is worth special notice. 3 x 7 21 Ex. 1. 221 72 172 7. It is much easier to find approximately the value of 121, and divide the result by 7, than to find the values of 13 and /7, and divide the former by the latter. wy Ex. 2. Reduce to its simplest form 6 (6 - c)2 6 с ху = 22 4 Ex. 3. Find the arithmetical value of 2 - 13 The denominator is the difference of two quantities, one of hich is a quadratic surd. Now, we know that (2 – 13) (2 + 13) (13) = 4 - 3 1, and hence we see that by multiplying numerator and denominator by the sum of the quantities in the denominator we can obtain the denominator in a rational form. 4 4 (2 + 13) 4 (2 + 13) Thus, 2 3 (2 13) (2 + 13) 22 - (13) 2 4 (2 + 13) 4 (2 + 13) 4 (2 + 13) 1 = 14.92820. 4(472 – 3 13) na so, 412 + 3 13 (4 + 3 13) (4/2 - 3/3) 4(4 72 - 3 13) 4 (4 12 – 3 3) * (472 - 3/3). (4 m2)2 – (3/3) 32 27 We shall now give an example when the surds are not quadratic. Ex 4. Rationalize the denominator of x 1 Since (zł)2 – (94)12 is (Art. 29, page 175) divisible by at – y, it follows that the rationalizing factor is their quotient, which is easily found. 16. Surds may be reduced to a common index. Ex. 1. Express ma and 1/6 as surds having a common index. Since ma am, and */b bn, it follows that, by reducing the fractional indices to a common denominator, the given surds become respectively amn, bmn, or "m/a", mm/6m. yš 1 1 |