Multiplying these powers, in their order, by the 4th, we shal obtain the 5th, 6th, 7th and 8th, the same as the 1st, 2d, 3d, and 4th; and so on.

266. The common rules for multiplication and division of radicals will apply to imaginary quantities, with a simple modification respecting the law of signs.

Let it be required to find the product of Va and V—b.

To obtain the true result, we must separate the imaginary symbol -1 from each factor.

Thus

But if we multiply by the common rule for radicals, (253), we

shall have

v=axv=b = √(—a) · (—b) =√ ab,

a result erroneous with respect to the sign before the radical. Proceeding as in the first operation we find that

Thus, like signs produce, and unlike signs produce +. Hence, 1.- The product of two imaginary terms will be real, and the sign before the radical will be determined by the common rule reversed.

We may operate in like manner in division of imaginary quantities. Thus,

That is, like signs produce + and unlike signs produce. Hence,

2. The quotient of one imaginary term divided by another will be real, and the sign before the radical will be governed by the common rule.

267. Let us assume the equation

in which a and a' are real. By transposition,

a-a' = (b'—b)√—1.

Now it is evident that in this equation

a = a'.

(2)

For, if a> a' or a <a', then the first member of equation (2) is different from zero, and real. But this can not be, because the second member is either nothing or imaginary. Hence a = a'; and equation (2) becomes

0 = (b'—b)√—1,

which can only be satisfied by putting

b = b'.

Hence,

If two imaginary quantities are equal, then the real parts are equal, and the coefficients of the imaginary symbol are also equal. 268. These principles may now be applied in the following

6. Multiply a +V—c by V—a+√/c. Ans. (a+c)√=1.

7. Divide 91-10 by 3√—2.

Ans. 31/5. 16

Ans.

a Nã

C

12. Find the values of x and y in the equation a+y+x√—c

=c+x+y√ —a.

Ans.

x = a+√ uc;

y=c+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 112 is a surd, for V12=2/3. 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, Va+26 is a surd, considered as an algebraic expression. But if a = 13 and 6, we have Va+26 √ 13+12=√‍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 a/b and cd be the two surds, reduced to their simplest form. Their product will be

acvbd.

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 not. But this factor must be irrational, otherwise the given surds

are not in their simplest form. Therefore the product acv bd, 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 a and b be the two surds; and, if possible, suppose

c being rational. Squaring both members, and transposing a+b, 2V ab = c2—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 ya+v/b=v/c, in which c is rational, but Vca surd.

Squaring both members, and transposing,

2V abc-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+by/x=c+dvy,

the surds being in their simplest form. By transposition,

b√x—dv/y = c—u.

(1)

(2)

Since the second member is rational, equation (2) can not be true if the surds have not the same irrational part; (2). Therefore √xy, and the equation may be written,

(b-d)v/xc-a,

which can be true only when b—d — 0 and c—a=

=

(3)

= 0; for other

wise, we should have a surd equal to a rational quantity or to zero: a=c, and b√x=dvy.

Hence, in (1),

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 √7—√2 are binomial surds. 274. If we square a binomial surd in the form of a ± √/b or Vab, the result will be a binomial surd. Thus,

(3+√/5)2 = 14+6√/5;

(√7-1/2)'=9-21/14.

Hence, a binomial surd in the form of a ±√b 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 ±√/b, let us assume

in which one or both of the terms in the first member must be irrational, because the second member is a surd.

Subtracting (4) from (3), and then taking the square root of the result,

Now it is obvious from these equations that x and y will be rational when ɑ2—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: