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.... (1.)

C + y = a .......................... (3.),

and 2 w wy = Wb or 4 xy = 6............... (4.) From (3) and (4) we easily find x = { (a + N a* - b),

and y = } (a - Naš - b). Hence, from (1), the square root required is— Vi (a + Wat ) + V1 (a - - 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 w14 + 6/5 = Na + Ny.
Squaring, then, 14 + 65 = a + y + 21 xy.
Hence, equating the rational and irrational parts

2 + y = 14.......... ............. (2.),

21xy = 615 or 4 xy = 180 ................ (3). From (2) and (3) we easily find x = 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 + 1496 = + Ny. Squaring, &c., we have, x + y = 39 ;

and 4 xy = 1496. From these equations we easily find x = 22, y = 17. Hence, the square root required is w22 + 117.

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

Ex. 1. Find the square root of 19 + 8 w3. We shall throw this expression into the form aż + 2 ab + 6, which we know is a perfect square.

Dividing the irrational term by 2, we have 4 13. 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.

Thus, we have 19 + 8N3 = (4) + 2 (4) N3 + (W3)2

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

Ex. 2. Find the square root of 29 + 1225.
We have 29 + 12 15 = (3)2 + 2 (3)2 V5 + (2/5)

= (3 + 2 25) The square root is therefore 3 + 2 25.

Ex. VI. Express with fractional indices1. Van Yaob, frj, Ya*b*. 2 By Need Mom : - Jab Jæy' m/a* Reduce to entire surds— 3. 3 13, 4 NZ, * V16, 3 I 5. 4. 4. 21, 9.3-1, 4.2-1, 1 (1)– 1. 5. 3 Vab, a Joe, (a + a), a = x Voordo, Na la TiNTa + 2)2 Reduce to a common index6. 32, v3.

7. 92, 93. 8. 212, 375.

9. Wa, WC. 10. (a + x)}, Ja - 7. 11. ai +1, be – .

Simplify12. /12, 948, 3 128, 1 9648. 13. V£a* + 4a"b, Taob* + B", 3

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Find the value of, 16. JĪ2 + 148 - 2/3, 156 + 189. 17. 1474- Novi + porto 18. 327 q** +6 - Ba**** + 3 364 a. Multiply19. a + vab + b by Va - Jo, at + b} by - Jo. 20. (x + y)2 by (x + y)š, a + b Vd by a - ab + bd. 21. ovo sū by a Va (a + b) - Vo. 22. + b + c + d} by až - 63 + ci - at

Divide 23. a* + xy + y by s + ały + y. 24. ad – ý by act + y$ Rationalize the denominators of

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1 . 0 .. 28. xt - g' V2 + 13' x + xłyt + i;

Find the square roots of 29. 11 + 4 N7, 8 + 2 Vī5, 30 – 10 15. 30. 8 + 2 N12, 9 – 6 /2, 20 – 10 13:

CHAPTER V.

RATIO AND PROPORTION.

Ratio.

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: 6 or be the ratio, and let each of its terms be increased by m. It will then become * *

a + ma Now, z im as (a + m) 6 = (b + m)d, or, as ab + bm = ab + am; or, as bmz am, or as b = d.

Hence the ratio is increased when 6 ta, that is, wheri it is a ratio of less inequality; and is diminished when 6 < d, 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 to oc : a or be the ratio, where x is small compared with a

mm (a + ac) am + 2 ax + acabat

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nearly =

(a + c) 3 a3 + 3 aʻac + 3 ax + 2013 32

· =1 +

a3 a + 32

nearly; and so on. Ex. (1002)2 : (1000)2 = 1004 : 1000 nearly.

(1002) : (1000)3 = 1006 : 1000 nearly.

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:b::c:d,

a c

or o = 2 is called a proportion. 26. The following results are easily obtained :

a b c b a 6 (1.) Since = as then ở x = 8 x or t =

::a:0:: 6:0 (alternando). (2.) 1 = 1 å or a

ib:a:: d:c (invertendo). Also, by Art. 64, page 214, we have (3.) a + 6:6:: 0 + d:d (componendo). (4.) a 6:6:: 0 - d:d (dividendo). (5.) a - 6:a:: 0 - dic (convertendo). (6.) a + b : a b :: C + d:c - d (componendo and

dividendo). 27. If a :b::c:d and e:f::g:h, we may compound the proportions. Thus we have n = 8 -....(1), and = .......(2).

(1) * (2), then, auf

or ae : bf :: cg : dh.

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