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= a...........

and y

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2 ay

2 + y

(3.), and 2 way or 4 xy == 6...............(4.) From (3) and (4) we easily find x = + (a + Wa? - b),

} (a

Na2 6). Hence, from (1), the square root required is, di (a + Na2 - 6) + N = (a - Na2 = 6).

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 214 + 675 N2 + N y ............

... (1.) Squaring, then, 14 + 6/5 = x + y + 2xy. Hence, equating the rational and irrational parts,

X + Y
14..........

.(2.), 625 or 4 xy

180......... (3). From (2) and (3) we easily find a = 9, y 5. Hence the square root required is 19 + 15 or 3 + 15. Ex. 2. Find the square root of 39 + N 1496. Let 39 + 7 1496

dă+ dý. Squaring, &c., we have, x + y = 39;

and 4 xy 1496. From these equations we easily find x =

17. Hence, the square root required is w22 + 717.

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 a2 + 2 ab + 69, 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.

22, y

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

(13)*

= (4 + 3)? The square root is therefore 4 + 13.

Ex. 2. Find the square root of 29 + 1275.
We have 29 + 12 15 (3) + 2 (3)2 5 + (25)

(3 + 2 V5)?
The
square

root is therefore 3 + 2 V5.

Ex. VI.

3

a

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Express with fractional indices 1. Não, Val?, Vizcay, Ya?b4.

xy ved om 2.

Jab Tay man
Reduce to entire surds—
3. 3 13, 4 N, 115, 3 VI.
4. 4. 21, 9.3-5, 4.2-4, 1 (1)– 1.

bc
a (a +
a

(a + c)?
Reduce to a common index-
6. 92, 3.

7. ^2, 13. 8. 2 V2, 3 35.

9. ma, r/5. 10. (a + c)}, Ta

ait, bë Simplify12. 712, 748, 3 128, 1 7648.

6 13. 14a3 + 4a2b, 9a378 + 70,

, B"Ja

X.

11. AP

a +

64 a

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Find the value of 16. /12 + 148 2 13, 156 + 1189. a*x*y;

axc

a

21. C

18. 727 am +673

Batm +8 + 3 64 am. Multiply19. a + Jab + b by Vā - No, at + 63 by - Vo. 20. (x + y)} by (+ y)}, a + b Nd by a? ab + bid. No

+
by

)

. a + 6 22. ab + b + ct + at by a 65 + cf. dt.

Divide23. 22 + xy + y2 by x + aty + y. 24. 28 – y by ct + yt. Rationalize the denominators of

4 25.

5

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3
4

1
26.
2 + 73 32

3 3

2

13 + 12 27. 1 + V2 + V2 + 13 + 13

N2 1

6 27 yt' + 73' x + atyt + y Find the square roots of29. 11 + 477, 8 + 2 Vī5, 30 - 10 V5. 30. 8 + 2 /12, 9 - 6 12, 20 – 10 73.

CHAPTER V.

RATIO AND PROPORTION.

Ratio.

a

a

a

22. The student is referred to Chapter II. of the Arithmetic section of this work for definitions and observations which need not be repeated here.

23. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by increasing the terms of the ratio by the same quantity. Let a :b or 2 be the ratio, and let each of its terms be

a + m increased by m. It will then become

b + m a + m Now,

= b + m

as (a + m) 6 = (b + m) a,

ō or, as ab + bm + ab + am; or, as bm = am, or as 6 = a. Hence the ratio is increased when b > a, that is, when

ъ it is a ratio of less inequality; and is diminished when 6 < a, that is, when it is a ratio of greater inequality.

Cor. It may be shown in the same way that

A ratio of greater inequality is increased, and a ratio of less inequality is diminished, by diminishing the terms of the ratio by the same quantity.

24. When the difference between the antecedent and consequent is small compared with either, the ratio of the higher powers of the terms is found by doubling, trebling, &c., their difference.

Let a + : a or be the ratio, where a is small compared with a. (a + x) az + 2 ax + cca

2 x Then

1 + nearly aa

a a + 2 x

nearly.

a + x

a

=

a

a

320

nearly

(a + x) as + 3 a-x + 3 axa + c3

=l+ as

as a + 3

nearly; and so on. Ex. (1002) : (1000) = 1004 : 1000 nearly.

(1002) : (1000)* = 1006 : 1000 nearly.

a

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or

a

a

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a or

с

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с

a

or

Proportion. 25. Proportion, as has been already said, is the relation of equality expressed between ratios. Thus, the expression a : b = c:d,

or a : 6°:: C:d,

a

7 ď
is called a proportion.
26. The following results are easily obtained :-

6
6

6 (1.) Since then a Х

Х
.. a:0::6:d (alternando).

6 d
(2.) 1 ; 1;
7

à

.. b:a:: d: c (invertendo). Also, by Art. 64, page 214, we have (3.) a + 6:6:: c + d: d (componendo). (4.) a - 6:6:: 6-d:d (dividendo). (5.) a - b:a::c- dic (convertendo). (6.) a + b: a - b :: C + d:c- d (componendo and

dividenilo). 27. If a:6::c:d and e:f::g:h, we may compound the proportions. Thus we have

6
.(1), and

sh (2).

cg (1) x (2), then,

of

dh or ae : bf :: cg : dh.

a

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e

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ae

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