335. A surd fraction can be transformed into an equivalent expression with the surd part integral. 336. Surds which have not the same index can be transformed into equivalent surds which have; see Art. 327. For example, take √5 and 11, 5a=5* = √/53= /125, (11)* =11* = √(11)2 = $121. 337. We may notice an application of the preceding Article. Suppose we wish to know which is the greater, 5 or 3/11. When we have reduced them to the same index we see that the former is the greater, because 125 is greater than 121. 338. Surds are said to be similar when they have, or can be reduced to have, the same irrational factors. Thus 4/7 and 5/7 are similar surds; 5/2 and 4/16 are also similar surds, for 4 $/16=8/2. 339. To add or subtract similar surds, add or subtract their coefficients, and affix to the result the common irrational factor. 340. To multiply simple surds which have the same index, multiply separately the rational factors and the irrational factors. For example, 3√2 × √√3=3√√6; 4√5×7√6=28/30; 23/4×3/2=63/8=6×2=12. 341. To multiply simple surds which have not the same index, reduce them to equivalent surds which have the same index, and then proceed as before. For example; multiply 4/5 by 2 11. By Art. 336, √5=125, 11 = /121. Hence the required product is 8/(125 x 121), that is 8/15125. 342. The multiplication of compound surds is performed like the multiplication of compound algebraical expressions. For example, multiply 6/3-5/2 by 2/3+3/2. 343. Division by a simple surd is performed by a rule like that for multiplication by a simple surd; the result may be simplified by Art. 335. The student will observe that by the aid of Art. 335 the results are put in forms which are more convenient for numerical application; thus, if we have to find the approximate numerical value of 3/2÷4/3, the easiest method is to extract the square root of 6, and divide the result by 4. 344. The only case, of division by a compound surd which is of any importance is that in which the divisor is the sum or difference of two quadratic surds, that is, surds involving square roots. The division is practically effected by an important process which is called rationalising the denominator of a fraction. For example, take the fraction 4 ; if we multiply both numerator and denomi 5.√2+2√3' nator of this fraction by 5√2-2√3, the value of the fraction is not altered, while its denominator is made rational; 345. We shall now shew how to find the square root of a binomial expression, one of whose terms is a quadratic surd. Suppose, for example, that we require the square root of 7+4√3. Since (√x+ √y)2 = x+y+2 √ (xy), it is obvious that if we find values of x and y from x+y=7, and 2(xy)=4/3, then the square root of 7+4/3 will be √x+y. We may arrange the whole process thus. Suppose √(7+4√3) = √x+√Y; Since x+y=7 and x-y=1, we have x=4, y=3; XXXV. Ratio. 346. Ratio is the relation which one quantity bears to another with respect to magnitude, the comparison being made by considering what multiple, part, or parts, the first is of the second. Thus, for example, in comparing 6 with 3, we observe that 6 has a certain magnitude with respect to 3, which it contains twice; again, in comparing 6 with 2, we see that 6 has now a different relative magnitude, for it contains 2 three times; or 6 is greater when compared with 2 than it is when compared with 3. 347. The ratio of a to b is usually expressed by two points placed between them, thus, a:b; and the former is called the antecedent of the ratio, and the latter the conse quent of the ratio. 348. A ratio is measured by the fraction which has for its numerator the antecedent of the ratio, and for its denominator the consequent of the ratio. Thus the ratio of a to b is measured by then for shortness we may a say that the ratio of a to b is equal to or is. b b 349. Hence we may say that the ratio of a to b is equal 350. If the terms of a ratio be multiplied or divided by the same quantity the ratio is not altered. 351. We compare two or more ratios by reducing the fractions which measure these ratios to a common denominator. Thus, suppose one ratio to be that of a to b, |