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Ex. 2. Reduce Va36 and Nadya to a common index.

The least common denominator of the fractional indices of the given surds is 4 x 3 or 12.

Hence we proceed as follows: Yar

(a*b)*

(ab)* Wa63, Pour eyja = (*y*)} = (x*y*) 1's */*)* = 5x20y. When the student has had a little practice, the first two steps of each of the operations may be omitted.

17. Addition and subtraction of similar surds.

DEF. Similar surds are those which have the same irrational factors.

Ex. 1. Find the sum of V12, 5 727, - 2 275. We have

12 + 5 127 - 2 275

22 x 3 + 5 132 x 3 - 2 157 x 3 = 2 1 3 + 5 x 373 - 2 x 573

2 3 + 15 73 - 10/3

(2 + 15 10)/3 = 7 N3. Ex. 2. Simplifya+b + 2ab + 33

2ab2 + 73
2ab + 62 a + 2ab + 62
The given expression-
(a + 6)6

6)26
(a - b) (a + b)
a + b

6

a + b
NO
a + 6

6
(a + b)2 (a 6)

4 ab

. b) (a + b)

a 18. Multiplication and division of surds.

The following examples will best illustrate these operations:

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a

a

No

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a

No

(a

a

Ex. 1. Multiply a Nöyz by b. Næjøu.
We have, a Vöyz x 6 Væyou =

ab Væʻyx x xyér = ab *yʻuz = abx®y Vuz. Ex. 2. Multiply a Õ + cvā by a – N sod. Arranging as in the case of rational quantities, we have

adb + c d

sbd
a po + ac d

- ab - cd 16
(a? - cd) 7b + alc - b) Jd
Ex, 3. Divide a 16 by b Va.
We have,

1
a No. Na adab
b da oda. Na

bia

7 When the divisor is a compound quantity it will generally be the best to express the surds as quantities with fractional indices, and proceed as in ordinary division.

19.The square root of a rational quantity cannot be partly rational and partly irrational. If possible, let Ja = m + vo; then, squaring, a = m? + 2 m /6 + b; or, 2 m jo (m + 6);

(m3 + 6) or,

b);

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a

2 m

that is, an irrational quantity is equal to a rational quantity, which is absurd.

20. To find the square root of a binomial, one of whose terms is a quadratic surd.

Let a + ū be the binomial.
Assume va + VC √x + √y,...

..(1); then, squaring, a + 1 = x + y + 2Way,... .(2).

Equating the rational and irrational parts (Art. 19), we

.(3.),

and y

6).

14.....

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a + y = a...... and 2 Nay

or 4 xy = b............. From (3) and (4) we easily find x =

} (a + Nab),

} (a - Nas Hence, from (1), the square root required is— di Na (a + Wat – ) + 17 (a

a - Não

Na - b). NOTE.—It is evident that, unless (a? 6) is a perfect square, our result is more complicated than the original expression, and therefore the above method fails in that case.

Ex. 1. Find the square root of 14 + 615.
Let 714 + 65 W X + N Y........

(1.) Squaring, then, 14 + 65 = x + y + 2xy. Hence, equating the rational and irrational parts,

...(2.), or 4 xy 180.......

..(3). From (2) and (3) we easily find x

5. Hence the square root required is 9 + 75 or 3 + 15. Ex. 2. Find the square root of 39 + N1496. Let 39 + 71496 =

dx + dý. Squaring, &c., we have, a + y = 39;

1496. From these equations we easily find x =

17. Hence, the square root required is w22 + 17.

21. The square roots of quantities of this kind may often be found by inspection.

Ex. 1. Find the square root of 19 + 8 13.

We shall throw this expression into the form aș + 2 ab + 6*, which we know is a perfect square.

Dividing the irrational term by 2, we have 4 23. Now all we have to do is to break this up into two such factors that the sum of their squares shall be 19. The factors are evidently 4 and 13.

9, y

and 4 xy

22, y

Thus, we have 19 + 813 = (4)2 + 2 (4) N3 + (13)2

(4 + 73)2 The square root is therefore 4 + 73.

Ex. 2. Find the square root of 29 + 1215.
We have 29 + 12 /5 = (3)2 + 2 (3)2 75+ (

25)

(3 + 2 55) The square root is therefore 3 + 2 15.

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5. 3 Jāb, a , (a + 2)

Ji

(a + x) Reduce to a common index6. V2, 3.

7. V2, 33. 8. 2 V2, 395.

9. ma, . 10. (a + 20),

11. až +, bi-. Simplify12. 712, 748, 3 V28, 9648. 13. 14a3 + 4aʼl, Va383 + 76,

b

.

3

sla +

64 a

3

14.

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+3 as

C

1 2a

la? 3 a?
+ aʻ,

aʻx.
2:
2c3 + 1 ola

(x + a)? (a2

a)
+
9 m2

(.ac a) (c + a):

3 nc

3

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21. C

at 6

Find the value of 16. V12 + 148 - 2 V3, 156 + 7189.

a*x*yg 17.

Vab* b

b 18. 327 an +673 8a4m + 9 + 3 764 a".

Multiply19. a + Vab + b by No, at + bi by sa - No. 20. (x + y)} by (x + y)}, a + b Nd by a? ab + 6%d.

T.

by Ja (a + b)

NO 22. ax + bt +

+ b + ci + als by ai 63 + cat Divide23. a* + xy + y2 by x + ały + y. 24. a* - y by ał + yt. Rationalize the denominators of

4 25. N37'5 3 4

1 26. 2 + 73 372 - 2 13' 15 13 3

2

3 + 12 27. 1 + 2 + 13 12 + 13 + J5 13

05:03-08 1

6 28. xt - !' V2 + 73 x + ætyk + y

Find the square roots of — 29. 11 + 4 17,8 + 2 Vī5, 30 10 25. 30. 8 + 2 V12, 9 - 6 12, 20 - 10 13:

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