ON SURDS. (128.) When a magnitude or number cannot be expressed in finite terms without the help of a fractional index, it is called a SURD: thus the square root of 2, the cube root of 3, the nth root of a + b, the cube root of (a + x), &c. &c. may be expressed either by 2, 3, a + b, y(a + x)2, &c. or by 25, 33, (a+b), (a+x), &c. L Note. The precise value of these quantities cannot be ascertained; it can only be expressed by means of decimals or series which do not terminate; and in this sense they are called irrational, to distinguish them from all other quantities whatever, integral or fractional, whose values are determinate, and which are therefore denominated rational. Surds in their radical form, when properly reduced, are subject to all the ordinary Rules of Arithmetic. The Reduction of Surd quantities. CASE I. (129.) A rational quantity may be reduced to the form of a surd, by raising it to the power denoted by the root of 'he surd. Example:. 1. Reduce 3 to form of the square root, and it becomes 3* or √9. 3. Reduce a+b...... square root,.... √(a+b)2. (130.) Surd quantities of different indices are reduced to equivalent ones with the same index, by bringing their frac tional indices to a common denominator. 3 16 Examples. 1. Reduce at and at to surds of the same index. 2 and 6 1 2 1 3 : a2= a2= a2, } common denominator, are which are surds with the same 6 index ✔. 2. Reduce 35 and 5t to surds of the same index. The fractions and reduced to a common denominator, are 2 1 3 2 2 6 6 6 6 Now 3=3=81; and 5=5°=√125. Ex. 3. Reduce a and b Ex. 4. Reduce cand d I Ex. 5. Reduce 33/2&25 Ex. 6. Reduce 41 and 15 to Surds 15 Ans. Ja and √b. with the Ans. c and Hd2. same index. CASE III. 6 6 (131.) Surd quantities are reduced to their simplest form, by observing whether the quantity under the radical sign contains a power corresponding to the given surd root, and then extracting that root. Note. The quantity without the radical sign is called the co-efficient of the surd; and it is evident, that this quantity may always be put under the radical sign, bu raising it the power denoted by the index of the Thus, 7a2x=(by Case I.) √7ax7ax √2x. #49a2 × √2x= √98ax. Also, x2a-x= x2× √2a-x. =x2x (2a-x) = √2ax2-x. CASE IV. (132.) If the quantity under the radical sign be a fraction, it may be reduced to an integral form by the following Rule. Multiply the numerator and the denominator of the fraction by such a quantity as will make the denominator a complete power, corresponding to the root; then extract the root of the fraction whose numerator and denominator are complete powers, and take it from under the radical sign. The Fundamental Rules of Arithmetic applied to (133.) The Addition and Subtraction of Surd Quantities. Rule. Reduce the quantities to their simplest form; and if the surd part be the same in both, then their sum or difference will be found by taking the sum or difference of their co-efficients, and annexing the common surd to the result. .. the sum =4a√x+2a/x=(4+2a) × √x=6a/x. the difference =4a/x-2a/x=(4a-2a) x/x=2a√x. are 2. Find the sum and difference of 192 and 24. By Case III.192=64 × 3 = 43/3, and 24= 8 x 3 = 23; ..192+24=(4+2) × √3=63/3 or 233. 3. Find the sum and difference of 8 The two fractions and 27 48 and 27 162 162 1 reduced to a common denominator, Note. If the surd part is not the same in the quantities which are to be added or subtracted from each other, it is evident that the addition or subtraction can only be performed by placing the signs + or - between them. 4. Add27ax and √3a2x together.......... 5...../128 and 172 6.....135 and 40 Ans. 4a 3x. .. 14/2. ..55 9. Required the sum of 248 and 93108....Ans. 8√3+273/4. 10. Find the difference of and... 11. Required the difference of 12x2y and √27y. ... Ans. O. Ans. (2xy+3y2).3. (134.) The Multiplication and Division of Surd quantities. Rule. Reduce the quantities to equivalent ones with the same index, and then multiply or divide both the rational and the irrational parts by each other respectively. By reduction, 2/3=2x3=2x3=227, 3 |