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expression, with the sign of one of its terms changed; and repeat the operation in the same way, as long as there are surds, when the last result will be rational.

2. When the terms of the binomial surd are odd roots, the rule becomes more complicated; but for the sum or difference of two cube roots, which is one of the most useful cases, the multiplier will be a trinomial surd, consisting of the squares of the two given terms and their product, with its sign changed.

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sign of the binomial, and the lower with the lower; and the series continued to n terms.

This multiplier may be derived from observing the quotient which arises from the actual division of the numerator by the denominator of the following fractions: thus,

ya I.

23–1+2n-2y + 2n=3y2+, &c., ..... tyn–1 to n terms, x+y whether 'n be even or odd.

y” II.

-27-2y+X-32-, &c., -Yn-1 to no 2十

ty terms, where n is an even number.

2. tyn III.

=y--1 Xn-t-a-342--, &c., +yn-1 to n x+y terms, when n is an odd number. Now, let xn = a, YN =b; then & Wij a, y="/b, and these fractions

0
b

ato severally become

and nga-ngingatni

nih And, since 2

xn--2 Van-, &c., also y2 = n/b2, 33 ="/b3, &c., therefore,

En yan-jan3b2

14 nyan_25+ njan312+, &c. ...+ n/2n-1 nya

no to n terms; where n may be any whole number whatever. And, ato

an-1na+"/ mj an2+. n; an-312 ---, &c.,

·="bnto n terms;. where the terms b and my bn-i liave the sign t, when n is an odd number; and the sign --, when n is an even number.

Now, since the divisor multiplied by the quotient gives the dividend, it appears from the foregoing operations, that, if a binomial surd of the form n/a-nyhbe muitiplied by / an-ib njan-26+, &c.,...+nyon-1, (n being any whole number whatever), the product will be a

b, arational quantity; and if a binomial surd of the form "ya+wybbe mubtiplied by "jan1 Jan-25 to niyan-b2 &c.,

En n-1, the product will be a tb, or ,~- -b; according as the index n is an odd or an even number. See my Elementary Treatise on Algebra, Theoretical and Practical.Ed.

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EXAMPLES.

1. To find a multiplier that shall render 5 + 73 rational.

Given surd 5 +13
Multiplier 5 - V3

Product 25 - 3= 22, as required. 2. To find a multiplier that shall make v 5 + V3 rational

Given surd v5 tv3
Multiplier V5 – V3

Product 5 -- 3 = 2, as required. 3. To find multipliers that shall make 4/5 + 4/3 rational

Given surd 4/5 + 4/3
1st multiplier V 5 - V3

1st product ✓5- V3
2d multiplier V5+ V3

2d product 5 -- 3 = 2, as required. 4. To find a multiplier that shall make V7+ 33 rational.

Given surd »7 + V3
Multiplier V 72 — 7 (7 X 3) +V 39

7+y (3 X 70)
- V 13 x 79)-V (7 X 32)

+ 3(7 X 34) + 3

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Product 19 +3= 10, as was required. 5. To find a multiplier that shall make v 5 -V x rational.

Ans. V5 to v 6. To find a multiplier that shall make vat v b rational.

Ans. Na 7. To find a multiplier that shall make a + b rational.

Ans. Q. 8. It is required to find a multiplier that shall make 1-32a rational.

Ans. 1 + 2a +} 40. 9. It is required to find a multiplier that shall make 3. — *V2 rational..

Ans. V9+164-4V 4. 10. It is required to find a multiplier that shall make » (a") + V (6%), or as + rational.

Ans 1/(a*b*) + V (a%). - be.

CASE XIV.

To reduce a fraction, whose denominator is either a simple or

a compound surd, to another that shall have a rational denominator.

5 RULE.-1.-When any simple fraction is of the form

va' multiply each of its terms by Va, and the resulting fraction

by a will be

a

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Or, when it is of the form

multiply them by V a , and k3 am the result will be

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2. If it be a compound surd, find sucn a multiplier, by the last rule, as will make the denominator rational; and multiply both the numerator and denominator by it, and the result will be the fraction required.

EXAMPLES

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V3 6V 53

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3 1. Reduce the fractions and to others that shall

✓ 3 ŠV 5 have rational denominators. 2 2 V3 2 V3

3

3 Here Х

and V3

3 34 53

6

4/ 125 the answer required. 2 X 5 5

5

3 2. Reduce

to a fraction whose denominator

V5-V2 shall be rational. 3

✓ 5+2 3 5+32 3V5+3V2 Here

Х
V5-V 2 ✓ 5 + 2 5 - 2

3
5 + 2

5 + 2 the answer required. 1

V 2 3. Reduce

to a fraction, whose denominator shall

3 -- V 2 be rational.

m*

V2

9.-2

7

V2 X (3 +12) 3V2 +2 27-31 Here 3- V2 (3 - V2) X (3 + V2)

2 3
7 +5v2 the answer required.

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4. Reduce

to a fraction that shall have a ra✓7+V3

(42) - V (18) tional denominator.

Ans.

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5. Reduce

to a fraction that shall have a rational 3 + *

320 XV 2 denominator.

Ans.

9 VE 6. Reduce

to a fraction, the denominator of which atvo

ata to b - 2avb shall be rational.

Ans.

a? - b 10 7. Reduce

to a fraction that shall have a ra

2/7 375 tional denominator. Ans. 5x [% (49) +(35) +3/(25)]

3 8. Reduce

to a fraction that shall have a raV 9 + 10 tional denominator.

3V9 33/(10) + (300) Ans.

19 4 9. Reduce

to a fraction that shall have a ra.

44+5 tional denominator.

9. Reduce v4+ v5

Ans. 4 {-V10 – 2 V2 +(2+ v 5)

(5) xv5}.

OF ARITHMETICAL PROPORTION AND

PROGRESSION.

ARITHMETICAL PRORORTION, is the relation which two quantities of the same kind, have to two others, when the difference of the first pair is equal to that of the second.

Hence, three quantities are said to be in arithmetical proportion, when the difference of the first and second is equal to the difference of the second and third.

Thus, 2, 4, 6, and a, a +b, a + 2b, are quantities in arithmetical proportion.

And four quantities are said to be in arithmetical proportion, when the difference of the first and second is equal to the difference of the third and fourth.

Thus, 3, 7, 12, 16, and a, at b,c,c + v, are quantities in arithmetical proportion,

ARITHMETICAL PROGRESSION, is when a series of quantities increase or decrease by the same common difference.

Thus, 1, 3, 5, 7, 9, &c., and a, a to do a + 2d, a + 3d, &c., are increasing series in arithmetical progression, the common differences of which are 2 and d. And 15, 12, 9, 6, &c., and a, a

d, a 2d, a

3d, &c., are decreasing series in arithmetical progression, the common differences of which are 3 and d.

The most useful properties of arithmetical proportion and progression are contained in the following theorems :

1. If four quantities are in arithmetical proportion, the sum of the two extremes will be equal to the sum of the two

means.

Thus, if the proportionals be 2, 5, 7, 10, or a, b, c, d, then will 2 +-10 = 5 + 7, and a +d=b + c.

2. And if three quantities be in arithmetical proportion, the sum of the two extremes will be double the mean.

Thus, if the proportionals be 3,6,9, or a, b, c, then will 3 +9= 2 X 6 = 12, and a to= 2b.

3. Hence an arithmetical mean between any two quantities is equal to half the sum of those quantities.

2 +4 Thus, an arithmetical mean between 2 and 4 is

2

5 +6 = 3; and between 5 and 6 it is

5. 2

atb And an arithmetical mean between a and b is

2 4. In any continued arithmetical progression, the sum of the two extremes is equal to the sum of any two terms that are equally distant from them, or to double the middle term, when the number of terms is odd.

*

a+20

* If two or more arithmetical means between any two quantities be required, they may be expressed as below.

2a +6 Thus, and

two arithmetical means between a and b, 3

3 a being the less extreme and b the greater,

na tb (n-1)a +-2b (n--2) a +36 atno And

&c. to

- any numn+1 n+1

nt-1

b ber: (n) of arithmetical means between a and b; where

is the common difference; which, being added to a, gives the first of these means; and then again to this last, gives the second ; and so on.

n+1'

a

n+1

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