CHAP. XXIX. ON THE REDUCTION OF SURDS. 244. SINCE it has been shown in the preceding Chapter that any rational quantity may be reduced to the form of a surd, we have now to consider how different surds, as a and /a, may be reduced, so that the radical sign may be the same in each. For this purpose, it is necessary to bear in mind, first, that these surds may be represented by fractional indices, and then to consider attentively the properties of fractions, as pointed out in Chapter X. Representing then these surds by their fractional indices, they become a1 and a*; and since we know that is equiva lent to, and to, we see at once that and a are respectively equivalent to a and a; and again that these quantities may be represented by 12/at and a3, of which the radical sign is the same in both. Hence arises the following general rule : Surds with different indices may be reduced to equivalent ones having the same radical sign, by reducing their fractional indices to a common denominator. If therefore it be required to reduce a and with the same radical sign, we have only to reduce fractions with a common denominator, viz. /a3 and a will be the surds required. a to surds and to and g, and 245. There is however another case to be considered, viz. the reduction of surds to their simplest form. In the consideration of this subject, it will be necessary to revert to Chap. XIV., in which irrational numbers, as derived from square square roots, have been already treated of, and in which it has been shown that such irrational numbers may be in many cases expressed in simpler forms by representing those numbers by their factors, and by extracting the root of any one of those factors which might be a complete square, and placing it before the radical sign, as in the instance of √8, which being equivalent to √4 × √2, may be more simply expressed by 22. The method, at which we have now arrived, of expressing surds by fractional indices, will enable us to apply the same principles generally to surds of every description. If, for example, it were required to express in a simpler form the following surd, viz. 2/a3, represented by its fractional index as a*, we shall first consider this quantity as product of two powers of a, and recollecting that the powers of any quantity are multiplied together by the addition of their indices, we shall immediately see that is the sum of 4 and 1, and consequently that a is the product of of a multiplied by a*, or a2, and may therefore be expressed by a2.a. Again a 3 being in like manner the product of a3 into a3, may be expressed a3. 3. So also if it were required to express in its simplest form, 3/108, we see at once that it is the product of 3/27 × 3/4, and by extracting the cube root of 27, we may therefore express it by 33/4. 246. The same principles will also apply to fractions, which may thus in many cases be reduced to an integral form. For example, we have already seen that the fraction 1 may be represented as at. If it were required therefore to reduce a2 to a simpler form the fraction- we shall first consider this Vas' fraction as the product of a2 multiplied into 1 39 and since we know know that this last fraction is equivalent to a, we have only to multiply a2 into a. Reducing the indices therefore to a common denominator, they will become 4-44. Consequently is equivalent to a*. This last surd being again /a3 a2 the product of a×a may be lastly expressed by a. ŵ/a. Again, if it were required to reduce 3/16 to an integral 3 8 x 2 ; 27 × 3 form, by dividing it into its factors, it becomes and by extracting the cube roots of 8 and 27, we obtain If the quantity under the radical sign be a fraction, transform it into an equivalent fraction whose denominator shall be a complete power corresponding to the root, extract the root of that complete power, and take it from under the radical sign. |