19. Extract the square root of 256x13 – 512x+640x3 – 512x3 + 304 - 128x ̄3 + 40x ̄ α 20. If a' = ba, shew that (1)=ai~'; and if a = 2b, shew that b=2. XIX. SURDS. 6 279. When a root of an Algebraical quantity which is required, cannot be exactly obtained, it is called an irrational or surd quantity. Thus /a or a3 is called a surd. But 3/aR or a3, though apparently in a surd form, can be expressed by a2, and so is not called a surd. The rules for operations with surds follow from the propositions established in the preceding chapter, as will now be seen. 280. A rational quantity may be expressed in the form of a given surd, by raising it to the power whose root the surd expresses, and affixing the radical sign. = Thus a = √a2 Ja3, &c.; and a +x=(a+x)". In the same manner the form of any surd may be altered; thus The quantities are here raised to certain powers, and the roots of those powers are again taken, so that the values of the quantities are not changed. 281. The coefficient of a surd may be introduced under the radical sign, by first reducing it to the form of the surd and then multiplying according to Art. 271. For example, a √x= √a2 × √x= √(a2x); ay3 = (a2y3)§; x √ (2a − x) = √(2ax2 — x3) ; a × (a − x)3 = {a2 (a — x)313 ; 4 √2 = √(16 × 2) = √32. 282. Conversely, any quantity may be made the coefficient of a surd, if every part under the sign be divided by the quantity raised to the power whose root the sign expresses. Thus √(a2 — ax) = a1× √(a−x); √(a3 - a2x) = α √(a− x) ; a (a2 — a3)* = aa × (1—~*)*; √/60 = √(4 × 15) = 2 √/15 ; a 283. When surds have the same irrational part, their sum or difference is found by affixing to that irrational part the sum or difference of their coefficients. Thus a√xb√x=(a+b) √x; √300±5 √3 = 10 √3 ± 5 √3 = 15 √3 or 5 √3; √(3a3b) + √(3x3b) = a √(3b) + x √(3b) = (a + x) √(3b). 284. If two surds have the same index, their product is found by taking the product of the quantities under the signs and retaining the common index. 1 1 Thus axb2= (ab), (Art. 267); √2 × √3 = √6; (a + b)1 × (a − b) 1 = (a2 — bo)§. 285. If the surds have coefficients, the product of these coefficients must be prefixed. Thus a√xxby=ab √(xy); 3/8 x 5 √2=15/16 = 60. 286. If the indices of two surds have a common denominator, let the quantities be raised to the powers expressed by their respective numerators, and their product may be found as before. Thus 2a × 3a = 81 × 34 = (24)* ; (a + x)a × (a − x)2 = {(a + x) (a − x)3¦§. 287. If the indices have not a common denominator, they may be transformed to others of the same value with a common denominator, and their product found as in Art. 286. Thus (a2 — xo)1 × (a − x)3 = (a3 — x2)‡ × (a − x)2 = {(a2 — x2) (a − x)o11 ; 288. If two surds have the same rational quantity under the radical signs, their product is found by making the sum of the indices the index of that quantity. 289. If the indices of two surds have a common denominator, the quotient of one divided by the other is obtained by raising them respectively to the powers expressed by the numerators of their indices, and extracting that root of the quotient which is expressed by the common denominator. 290. If the indices have not a common denominator, reduce them to others of the same value with a common denominator, and proceed as before. Thus (a2 — x3) 1 ÷ (a3 — x3)* = (a2 — x2)3 ÷ (a3 — x3) 3 = { (a2 — x®)3 } } 291. If the surds have the same rational quantity under the radical signs, their quotient is obtained by making the difference of the indices the index of that quantity. 292. It is sometimes useful to put a fraction which has a simple surd in its denominator into another form, by multiplying both numerator and denominator by a factor which will render the denominator rational. Thus, for example, If we wish to calculate numerically the approximate value of 2 it will be found less laborious to use the equivalent form √3 293. It is also easy to rationalise the denominator of a fraction when that denominator consists of two quadratic surds. 294. By two operations we may rationalise the denominator of a fraction when that denominator consists of three quadratic surds. For suppose the denominator to be a + √b + √c; first multiply both numerator and denominator by Ja+b-c, thus the denominator becomes a+b-c+2 (ab); then multiply both numerator and denominator by a+b-c-2 (ab), and we obtain a rational denominator, namely (a+b-c) - 4ab, that is, a2 + b2 + c2 - 2ab2bc - 2ca. 295. A factor may be found which will rationalise any bino mial. 1 1 1 (1) Suppose the binomial a+b. Put x=a, y = b; let n be the least common multiple of p and q ; then x" and y" are both rational. Now (x + y) (x2−1 − x′′−3y + x" ̄3y3 — ... ± y"−1) = x" ± y", — where the upper or lower sign must be taken according as n is odd Take, for example, a+b; here n = 6. Thus we have as a rationalising factor that is, that is, a3 — a2b3 + a3b3 — ab + a3b3 — b3 ̧ The rational product is xo — yo, that is, aa — b3, that is, a3 — b2. |