2 That is, a is one of three equal factors which produce a2. .. a2 = Va2. 1 1 1 In like manner, a" × a” × an to n factors equals 1 ... That is, a is one of n equal factors which produce a. In like manner, the meaning of a" is made clear. That is, a" is one of n equal factors which produce am. 218. From this last, the rule may be deduced that, in a fractional exponent, the numerator denotes a power and the denominator a root. Negative Exponents. In sections (71, 72), we proved the theorems ao 1, and = It remains only to extend this proof further to 219. That is (73), any quantity with a negative exponent is equal to the reciprocal of that quantity with an equal positive exponent. 220. The reciprocal of a quantity is 1 divided by that quantity. The reciprocal of a fraction is that fraction inverted. Hence a" is the reciprocal of a"; that is, a " .. a" × a" = 1. 1 = ; an = a an If in the quantity a"-" m>n, the result will be a positive exponent; but if n> m, the result will be otherwise, and we shall have the expression a (n—m) For, = am-n it will appear that quantities with fractional and negative exponents are simply another form of notation for quantities which we already had the means of representing. To show that (ab)"a"b", and (ab) "=- : 1 1 1 anbn Raising both members of the equation to the npth power, we have, xnp amp. mp m Taking the npth root of both members, we have x = a1p = an, K 1. Simplify (y). EXERCISE 75. 2. Multiply together a, b, a, b, and c2. 3. Simplify (a), (e) + b a 3m 4. Express as radicals a3, b3, c11, d3ï, x2, b1. 5. Express with fractional exponents: Va2, Vb1113, Væ3у1, Vxaу3. 7. Divide ab by a-b. 8. Multiply æ1 — æ‡ył + y2 by x2+x‡ył + y2. 16. Extract the fourth root: x ̄‡ — 4 x ̄3⁄4y3 + 6 x ̄‡y3 — 4x ̄}y} + y}. CHAPTER XVIII. RADICAL EXPRESSIONS. 221. All expressions affected with the radical sign are called Radicals. 222 Those expressions which are affected with the radical sign and of which the exact root cannot be obtained are called Surds or Irrational Quantities. 5 Thus at or as is called a surd, but Va or a3 is not a surd, as the exact root can be taken. Surds are similar which have the same surd factor. 223. A mixed surd is one which contains a rational and a surd factor. Thus 5√2 and b√ac are mixed surds. The rational factors of mixed surds are regarded as the coefficients of the surds. An Entire Surd is one which has no coefficient. Thus √10, √ab, √(a+b)1. |