472

as + Zac

a?

a7 72 12. (en - am) + (

= (2# + a ) (3* + 47) (* + , .....(* + a2"). Find the square roots of 13. a + 6 + 6 + 2 (alb+ ałck + b*cı). 14. 4 2 – 12 x + 17 ago – 12 cus + 4c. 15. a’6-1 - 4 ab-} - 80-13% + 4a-2 + 8.

Find the cube roots of — 16. 2* + 9 x + 6 x š – 99x2 - 42 25 + 441 aš - 343. 17. «øy-1 + 3x+y=+ 3 x?y-} + 1. 18. ab (1 + 3 a-$5} + 3 a-{b} + a-18) (ab-1 - 3 ašt – + 3a6-} + 1).

CHAPTER IV.

SURDS. 12. A surd quantity is one in which the root indicated cannot be denoted without the use of a fractional index. Thus, the following quantities are surds :

fac + y Va + ac

Since, from what has been explained in the last chapter, these quantities may be written thus

az, (a’ + 2?)?, (a? + 62 + c)
a}, (a? + 20°)?, (aż + b3 + c)", (a

(x + y){ (a + x)*

) ut it follows that surds may be dealt with exactly as we deal with their equivalent expressions with fractional indices.

It is evident that rational quantities may be put in the form of surds, and conversely, expressions which have the form of surds may sometimes be rational quantities. · Thus, a = (a)3 = Vab; and Va + 3 a2b + 3 ab? + 63 = V(a + b)3 = a + b.

13. A mixed quantity may be expressed as a surd. Thus, 3 75 = 333. 35 = 133 x 5 = 3/135, and so, « Wy = Waa. Wy = "x"y.

14. Conversely, a surd may be expressed as a mixed quantity, when the root of any factor can be obtained. Thus, 18 a362 = 19 a’62 2 a

= 19 aʼ62. 12 a = 3 ab 12 a.
And Maạ + b2)®zc*y= Na + b2)®c®y} xy?

- Ha? + 62)82c*y." Tæy = (a” + 6)xy Jæy. 15. Fractional surd expressions may be 80 expressed that the surd portion may be integral.

The process is called rationalizing the denominator, and is worth special notice. Ex. 1. B = 13x7 121 121

to N7N 72-= 172 = 7 It is much easier to find approximately the value of V21, and divide the result by 7, than to find the values of J3 and /7, and divide the former by the latter..

xy Ex. 2. Reduce to its simplest form

Xy fxy(b − c) Vxy(6 – c).
No ac N 76 – c) 6-0

Ex. 3. Find the arithmetical value of To

The denominator is the difference of two quantities, one of hich is a quadratic surd.

Now, we know that (2 – 13) (2 + 13) = 22 - (3) = 4 - 3 = 1, and hence we see that by multiplying numerator and denominator by the sum of the quantities in the denominator we can obtain the denominator in a rational form. 4

4(2 + 13) 4 (2 + 13) US, 2 – 23 = (2 – 13) (2 + 13) – 22 – ( 13ja

Thus, –

4

= 4(2 + 1.73205) = 4 (3.73205.)
= 14.92820.

4(472 – 3 13) na sos 4 12 + 3 13" (4 + 3 13) (4 12 - 3 /3) 4(4 JZ - 3 13) _ 4 (4 12 – 3 1/3) = 4(472 – 3 13). (4 1/2)2 – (3 13) 32 – 27

We shall now give an example when the surds are not quadratic.

Ex 4. Rationalize the denominator of I T

Since (ac+)12 - (y)12 is (Art. 29, page 175) divisible by at - y, it follows that the rationalizing factor is their quotient, which is easily found.

16. Surds may be reduced to a common index. Ex. 1. Express ma and */b as surds having a common

index.

67. it follows.

Since m'a = am, and /6 = bn, it follows that, by reducing the fractional indices to a common denominator, the given surds become respectively amn, bạm, or maya", mm/bm.

ti Def. Simion and subtrons may be omittice, the #

Ex. 2. Reduce a3 and Vaby 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:

Val = (a*b)} = (a*b)id = ab)3 = "al",

Szorja = (acky?)} = (c®y) is = "un*y)* = 5%20y%. 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 712, 5 227, - 2 275.
We have

112 + 5 127 – 2 N75
= 122 x 3 + 5/32 3 - 2 152 x 3
= 2 x 3 + 5 x 3 N3 – 2 x 573
= 2 x 3 + 15 N3 - 10 N 3

= (2 + 15 – 10) 3 = 7 N3. Ex. 2. Simplify

lab + 2ab2 + 63 Jab - 2ab2 + 13

Naž - 2ab + 32 Na + 2ab + b2 ° The given expression

- la + b)26 la - 6) 6

N (a - b)2 li a - b

+ b a - 6 a - Onora + E NO

la - b a + b _ (a + b)2 – (a - b)2, 4 ab Ji

(a - b) (a + b) No = až – 72 V. 18. Multiplication and division of surds.

The following examples will best illustrate these operations:

(a + b)?

a

+ b

Ex. 1. Multiply a Nærøyz by b. Næju.
We have, a Nxyz x 6 Nxyču = ab vxyz * ayêu

= ab Væ*fuz = aba?y? Vuz. Ex. 2. Multiply a Vī + cvā by a – Vod. Arranging as in the case of rational quantities, we have

a doted

a - Vod
a 1b + ac Jd

- ab Nă - ed Nb
Tak – cd) 1b + alc – b) ja
Ex. 3. Divide a so by 6 Jā.
We have, a vb a Vo.Na avab 1

ba ba. Na = b. a =ē Nab. 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 16 + b;

or, 2 m /b = a - (m? + b);
or, vb = a
- a - (m3 + 6).

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 + Jū be the binomial.

Assume vā + 16 = 1x + xy ........... ............ (1); then, squaring, a + wū = x + y + 2 Næy,............... (2).

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

the binomet 19.2W arty