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249. DEFINITION. If the root of a quantity cannot be exactly obtained, the indicated root is called a surd.
Thus V2, 35, Vas, Va? + b2 are surds.
By reference to the preceding chapter it will be seen that these are only cases of fractional indices; for the above quantities might be written
21, 53, al, (a?+62). Since surds may always be expressed as quantities with fractional indices they are subject to the same laws of combination as other algebraical symbols.
250. A surd is sometimes called an irrational quantity ; and quantities which are not surds are, for the sake of distinction, termed rational quantities.
251. The order of a surd is indicated by the root symbol, or surd index. Thus x, "a are respectively surds of the third and nth orders.
The surds of the most frequent occurrence are those of the second order; they are sometimes called quadratic surds. Thus V3, va, Vx+y are quadratic surds.
252. If the surd contains a factor whose root can be extracted, the indicated root is called a mixed surd.
This factor can evidently be removed and its root placed before the radical as a coefficient. It is called the rational factor, and the factor whose root cannot be extracted is called the irrational factor.
253. When the coefficient of the surd is unity, it is said to be entire.
254. When the irrational factor is integral, and all rational factors have been removed, the surd is in its simplest form.
255. When surds contain the same irrational factor they are said to be similar or like.
573, 2V3, V3 are like surds.
But 3 v2 and 2 v3 are unlike surds.
256. In the case of numerical surds such as V2, 95, although the exact value can never be found, it can be determined to any degree of accuracy by carrying the process of evolution far enough. Thus V5=2.236068...
; that is v5 lies between 2.23606 and 2.23607; and therefore the error in using either of these quantities instead of v5 is less than •00001. By taking the root to a greater number of decimal places we can approximate still nearer to the true value.
It thus appears that it will never be absolutely necessary to introduce surds into numerical work, which can always be carried on to a certain degree of accuracy; but we shall in the present chapter prove laws for combination of surd quantities which will enable us to work with symbols such as V2, 95, Va, with absolute accuracy so long as the symbols are kept in their surd form. Moreover it will be found that even where approximate numerical results are required, the work is considerably simplified and shortened by operating with surd symbols, and afterwards substituting numerical values, if necessary.
257. A surd of any order may be transformed into a surd of a different order having the same value. Such surds are said to be equivalent. Examples. (1) $2=2}=21s = 1/24
(2) parap=api=pg/a9. 258. Surds of different orders may therefore be transformed into surds of the same order. This order may be any common multiple of each of the given orders, but it is usually most convenient to choose the least common multiple.
Example. Express Vas, %62, 9a5 as surds of the same lowest order.
The least common multiple of 4, 3, 6 is 12; and expressing the given surds as surds of the twelfth order they become a', 19/08, alo.
259. Surds of different orders may be arranged according to magnitude by transforming them into surds of the same order.
Example. Arrange V3, 36, 10 according to magnitude.
The least common multiple of 2, 3, 4 is 12; and, expressing the given surds as surds of the twelfth order, we have
V3 = 1/36 = 1729,
10=103= 1000. Hence arranged in ascending order of magnitude the surds are
V3, vio, y6.
EXAMPLES XXXII. a.
Express as surds of the twelfth order with positive indices : 1. em.
3. Vari xa-x-2 1
- - امونه
Express as surds of the nth order with positive indices :
Express as surds of the same lowest order: 15. Ja, al. 16. Ja, Na. 17. Şx, W X6, 2%205. 18. 12x4, 9310 19. a3b4, Vab. 20. Vacs, a.24. 21. 15, yii, %13. 22. 48, 13, 36. 23. 32, 98, 94.
260. The root of any expression is equal to the product of the roots of the separate factors of the expression.
= a. 1. Similarly,
Vabc="a. Nb. Pe; and so for any number of factors.
Examples. (1) ^15 = V3 · V5.
(2) Şaøb= ya · bra” b.
(3) V50 = 25. v2=52. Hence it appears that a surd may sometimes be expressed as the product of a rational quantity and a surd; when the surd factor is integral and as small as possible, the surd is in its simplest form [Art. 254].
Thus the simplest form of v128 is 8 V2.
Conversely, the coefficient of a surd may be brought under the radical sign by first raising it to the power whose root the surd expresses, and then placing the product of this power and the surd factor under the radical sign. Examples. (1) 75= 49. V5= 245.
(2) a %= %a8. = asb. In this form a surd is said to be an entire surd [Art. 253].
261. To add and subtract like surds : reduce them to their simplest form, and prefix to their common irrational part the sum of the coefficients.
1 Example 1. The sum of 3 V20, 475,
15 =675+475+ mv
= Example 2. The sum of x V8x}a+y = y3a-2 73a
=x.2. Ya+y(-y) ya-2.z ya
= (2x2 - y2-24) ya. 262. Unlike surds cannot be collected.
Thus the sum of 5v2, -2v3 and V6 is 5V2 -2v3+ V6, and cannot be further simplified.
EXAMPLES XXXII. b. Express in the simplest form: 1. V288. 2. 147. 3. 256. 5. 3 150. 6. 27720. 7. 5245. 9. 3125. 10. V-2187. 11. 73623.
4. 432. 8. 1029. 12. 1279335.
13. - 108x4y? 14.35*y* +5. 15. 24+PyP. 16. Na3+2ab+ab. 17. 18.x*y – 24x3y2 +24x+y3 – 8xy*.
Express as entire surds: 18. 11 2. 19. 14 5. 20. 694. 21. 576. 4 77
2a 6+ 25.
xan + 1
1.C - y 31. (x+y) N x+y
- - x2
Na?x? Find the value of 33. 3 745 – 120+7 15. 34. 4703+5 7-8 728. 35. 44-5 /176+2/99. 36. 2/363 – 5 243+ 192. 37. 27189 +3875-7 56. 39.5 81 -7 192+4 648. 39. 3/162 - 7432 + 1250. 40.59-54-21-10+4 7686. 41. 4/128+4/75-5/162. 42. 5/24-254-76.
7 43. 1252 - 1294 – 48
3 V 3 263. To multiply two surds of the same order: multiply separately the rational factors and the irrational factors.
Examples. (1) 5/3x3/7=15 /21,
(2) 2/x x3 x=0x.