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Ans. _v3. -3 10. Expand (a+v=c): Ans. a -6a'c+c+(4a-4ac)
Vc. 11. Divide a’+v=a by a-V-a. Ans. a+1=a-1.
12. Find the values of x and y in the equation a+y+aV = c+x+yv-a.
S x = a +Vac;
PROPERTIES OF QUADRATIC SURDS.
269. A Quadratic Surd is the square root of an imperfect square.
270. The root of a number will be a surd, when the number contains one or more irrational factors. Thus V 12 is a surd, for V 12 =27/3. The surd factor 3, is called the irrational part of the given surd.
271. A quantity may be a surd when considered algebraically, even though its numerical value is rational. Thus, the quantity, Vä+26 is a surd, considered as an algebraic expression. But if a= 13 and b = 6, we have Va+20=V13+12=V 25 = 5, a rational quantity.
272. The following properties of surds are important both in a theoretical and a practical view. The radical expressions are supposed to represent irrational numbers.
1.- The product of two quadratic surds which have not the same irrational part, is irrational.
Let ab and cyd be the two surds, reduced to their simplest form. Their product will be
acid And since, by hypothesis, b and d are not the same numbers, one of them must contain at least a factor which the other does
But this factor must be irrational, otherwise the given surds
are not in their simplest form. Therefore the product acVod, is irrational; (270).
2.— The sum or difference of two quadratic surds which have not the same irrational part, can not be equal to a rational quantity. Let ya and be the two surds; and, if possible, suppose Va+yb = c,
(1) c being rational. Squaring both members, and transposing a+b, 2V ab = (–a–b.
(2) That is, we have an irrational quantity equal to a rational quantity, which is impossible. Therefore equation (1) cannot be true.
In like manner it can be shown that the difference of two surds, not having the same irrational part, can not be rational.
3.— The sum or difference of two quadratic surds which have not the same irrational part, can not be equal to another quadratic surd.
If possible, suppose Va+yb=Vc, in which c is rational, but V ca surd. Squaring both members, and transposing,
2V ab = c-a-b, which is impossible, because a surd can not be equal to a rational quantity
4.- In any equation which involves both rational quantities and quadratic surds, the rational parts on each side are equal, and also the irrational parts. Suppose we have a+byx=c+dy,
(1) the surds being in their simplest form. By transposition,
bvxdv/y =a. Since the second member is rational, equation (2) can not be true if the surds have not the same irrational part; (2). Therefore Vx=Vy, and the equation may be written, (6—d)/x = (–a,
(3) which can be true only when b-d
O and c-a=
0; for other wise, we should have a surd equal to a rational quantity or to zero: Hence, in (1), a=c, and by x=d1y.
SQUARE ROOT OF A BINOMIAL SURD.
273. A Binomial Surd is a binomial, one or both of whose terms are surds. Thus, 3+15 and 77-72 are binomial surds.
274. If we square a binomial surd in the form of a + Vb or Va+vb, the result will be a binomial surd. Thus,
(77–72)=9–2V 14. Hence, a binomial surd in the form of a ty may sometimes be a perfect square.
275. To obtain a rule for extracting the square root of a binomial surd in the form of a +7b, let us assume Vx+vy=Va+yb,
(1) in which one or both of the terms in the first member must be irrational, because the second member is a surd. Squaring both members, a+2Væy+y = aty 6.
(2) Hence from (272, 4) we have x+y = a,
(3) 2V xy = 16. Subtracting (4) from (3), and then taking the square root of the result,
VX-Vy=Vã–16. Multiplying (1) by (5),
x=y=va_b. Combining (3) and (6) we obtain
2 Now it is obvious from these equations that x and y will be rational when a--b is a perfect square. Moreover, the values of x and y in (7) and (8) will evidently satisfy equations (1) and (5). Hence, to obtain the square root of a binomial surd, we may proceed as follows:
Let a represent the rational part, and ✓b the radical part, and find the values of x and y in equations (7) and (8). Then if the binomial is in the form of atıb, as in equation (1), the required root will be
Vx+vy. But if the binomial is in the form of a b, as in equation (5), the required root will be
EXAMPLES FOR PRACTICE.
1. Required the square root of 7+473. In this example a = 7, and 1b = 413; or b=48. Hence,
7-V 49_48 y =
2 And we have
Vx+y= 2+1/3, Ans. 2. Required the square root of 11–815. In this example a = 11 and 1 b = 81-5, or b = -320. W. have, therefore,
11 +1 121 +320
= 16; 2
x - Vy=4-1-5, Ans.
2 and we have
Vx+vy= 2m+i'm-C, Ans.
4. What is the square root of 11+672? Ans. 3+1/2. 5. What is the square root of 7—413 ? Ans. 2-1/3. 6. What is the square root of 7—2710? Ans. 5-12 7. What is the square root of 94+4275? Ans. 7+375. 8. What is the square root of 28+1013? Ans. 5+1 3. 9. What is the square root of np+2m*—2mV np+mo?
Ans. V np+m'm.
10. What is the square root of bc+26V 10-2?
Ans. 6+1 bc-6. 11. What is the square root of 7+30V 2?
12. Find the value of V16+301-1+V16—301 1.
18. Find the value of V11+672+V7_27/10
. 14. Find the value of V314127=34 V144153.
15. Find the value of V17+127/2.
276. It is sometimes useful to transform a fraction whose denominator is a surd, in such a manner that the denominator shall become rational. The fraction is thus simplified, because, in general, its numerical value can be more readily calculated. This transformation is usually effected by multiplying both terms of the fraction by the same factor.