5. x2y-2+2+x-3y2 by xy1+x"1y. 6. a-4+a ̄2b-2+ba by a 7. x3y—3—x−3y3-3xy-1+3x-1y by xy-1-x-1y. 9. a3b¬3+a-3b3 by ab¬1+a-1b. 10. a ̄3+b−3+c-3-3a ̄1b ̄1c-1 by a1+b1+c1. 288. To shew that (ab)"a". br. (ab)"=ab.ab.ab...to n factors =(a.a.a... ton factors) × (b.b.b... to n factors) =an. br. We shall now give a series of Examples to introduce the various forms of combination of indices explained in this Chapter. EXAMPLES.-CVII. 1. Divide x3 — 4xy+4x3y+4y2 by x3+2x3y3+2y. 2. Simplify {(25)3. (29)933– ̧ 3. Simplify (2061)-3. 8. Multiply (a2+b3)3 by aa—b3. 9. Divide a-b by 3/a-3b. 10. Prove that (a2)m = (am)2. ~ 11. If am" = (am)", find m in terms of n. し 12. Simplify x+b+c+- 24-b+c xb+c-a \P 13. Simplify (****)" + (-)" ̄. 14. Divide 4a* by q-p ป 15. Simplify [{(a-") ̄"}"]÷[{(a")"}-P]. 16. Multiply aTM+bo−2c′′ by 2am —3b. 17. Multiply am-nb”-r by a”-mbr¬"c. clear 20. Multiply am―bam-1x+cam-2x2 by a2+ba"-1x- ca ̃ ̄2x2. 21. Divide x2rq-1-y2-1) by xq−1)+y¶p−1) ̧ 23. Multiply 3r + x2ry3 + x*y2+y3 by x2-yo. 24. Write down the values of 625 and 12-2. XXIV. ON SURDS. 289. ALL numbers which we cannot exactly determine because they are not multiples of a Primary or Subordinate Unit are called SURDS. 290. We shall confine our attention to those Surds which originate in the Extraction of roots where the results cannot be exhibited as whole or fractional numbers. For example, if we perform the operation of extracting the square root of 2, we obtain 1'4142..., and though we may carry on the process to any required extent, we shall never be able to stop at any particular point and to say that we have found the exact number which is equivalent to the Square Root of 2. 291. We can approximate to the real value of a surd by finding two numbers between which it lies, differing from each other by a fraction as small as we please. Thus, since √2=1'4142...... √2 lies between and which differ by 14 15 10 10' 10 And, generally, if we find the square root of 2 to n places of decimals, we shall find two numbers between which √2 lies, differing from each other by the fraction 1 10" 292. Next, we can always find a fraction differing from the real value of a surd by less than any assigned quantity. For example, suppose it required to find a fraction differ Now 2(12), that is 288, lies between (16) and (17), 293. Surds, though they cannot be expressed by whole or fractional numbers, are nevertheless numbers of which we may form an approximate idea, and we may make three assertions respecting them. (1) Surds may be compared so far as asserting that one is greater or less than another. Thus /3 is clearly greater than √2, and 3/9 is greater than 3/8. (2) Surds may be multiples of other surds: thus 2√2 is the double of √2. (3) Surds, when multiplied together, may produce as a result a whole or fractional number: thus 294. The symbols Ja, ja, Ja, Ja, in cases where the second, third, fourth and nth roots respectively of a cannot be exhibited as whole or fractional numbers, will represent surds of the second, third, fourth and nth order. These symbols we may, in accordance with the principles laid down in Chapter XXIII., replace by a, at, at, aa. απ. 295. Surds of the same order are those for which the root-symbol or surd-index is the same. Thus √a, 3√(3b), 4√(mn), r1 are surds of the same order. Like surds are those in which the same root-symbol or surd-index appears over the same quantity. Thus 2a, 3a, 4a are like surds. 296. A whole or fractional number may be expressed in the form of a surd, by raising the number to the power denoted by the order of the surd, and placing the result under the symbol of evolution that corresponds to the surd-index. 297. Surds of different orders may be transformed into surds of the same order by reducing the surd-indices to fractions with the same denominator. Thus we may transform x and y into surds of the same order, for and 3/x=x3 = x11 =12/x^, and thus both surds are transformed into surds of the twelfth order. EXAMPLES.-CVIII. Transform into surds of the same order: 1. x and y. 2. 4 and 5/2. 3. 4. /2 and /2. 5. n/a and "/b. 6. (18) and (50). 298. If a whole or fractional number be multiplied into a surd, the product will be represented by placing the multiplier and the multiplicand side by side with no sign, or with a dot (.) between them. Thus the product of 3 and 2 is represented by 32, |