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(136.) Scholium.-From the preceding rules we easily deduce the method of converting fractions, whose denominators are surd quantities, into others, whose denominators shall be rational. Thus, let
both the numerator and denominator of the fraction wo be multiplied
ANA by vx, and it becomes
; and by multiplying the numerator and
6 denominator of the fraction by y(a+x) or (a+x)}, it be
3a+x b(a+b)x_b(a+I)3. Or in general, if both the numerator
Va+x)' ata and denominator of a fraction of the form av de multiplied by 1.7o-s, it becomes a"
a fraction whose denominator is a rational
quantity. On the method of finding Multipliers which shall render Binomial
Surd Quantities Rational. (137.) Compound surd quantities are those which consist of two or more terms, some or all of which are irrational ; and if a quantity of this kind consist only of two terms, it is called a binomial surd. The rule for finding a multiplier which shall render a binomial surd quantity rational, is derived from observing the quotient which arises from the actual division of the numerator by the denominator of the following fractions. Thus,
- they + smsy + &c...+y*-to n terms, wheIy her n be even or odd.
2" -- y"
xn--"-by + masy? — &c. . -yn-' to n terms, when x+y n is an even number.
I'"+y" 3. =x*-1-2y+*-ye-&c. . +y- to n terms, when
x+y n is an odd number.* (138.) Now let xa=a, yn=b, then x=va, y=10, and these
ato fractions severally become
Parayat and Va+v0 by the application of the rules, already laid down, we have 2-= Van; zms=2>; za=s=-s, &c. ; also, ye=16*; yo=10°, &c. hence, zasy= Var-sxvv=Var26; zr—yo = Pan-sx V12= Va-362, &c. By substituting these values of sms, 21ay, 23yo, &c. in the several quotients we have
= Van-+ Van–11+ Dan-sb2+ &c.. +16=1 to n terms; Ža- vb
whole number whatever. And a+b =V2-16+ vcs6+ &c.. I 70i to n terms; Mato where the terms b and non- have the sign +, when n is an odd number; and the sign
when n is an even number. (139.) Since the divisor multiplied by the quotient gives the dividend, it appears from the foregoing operations that if a binomial surd of the form Da-o be multiplied by Var- + Var-+ van–32+&c.... + vb- (n being any whole number whatever), the product will be a—b, a rational quantity; and if a binomial surd of the form va+ vb be multiplied by va --Var-26+ 'va-32–&c... + vba-, the product will be
where n may
ty • For 1. =rty;
=x3+xy+xyz +ys, &c. 1-4
213ray+xyz-y3, &c. x+y
Erl M34+xay2-143+ys, &c. rty x+y
atb or a-b, according as the index n is an odd or an even number ; but here we inust observe, that the number of terms to be taken, is always equal to n.
Note.—The greatest use of this rule is, to convert fractions having surd denominators, into others which shall have rational ones.
Examples. 1. Reduce to a fraction which shall have a rational deno.
Now a-vr=va? – VX; put aʻ=c, then va?— WI=vc-vx. In this example n is equal to 2, and therefore the number of terms to be taken of the general series "Du"-1+ Van-26+&c. is 2, and because a:=c and l=x, the multiplier is c+ WX; whence /c+r
to a fraction which shall have a rational de
Here again n=2: therefore the number of the terms to be taken of the series, an--'-Van-b+ &c. is 2; and in this example, a=8 and l=3 : therefore the multiplier is V8-73; wherce
78-73 76 473-32
2 3. Reduce
to a fraction which shall have a rational deno
33- 3/2 minator.
Here 113, as3, and b=2; therefore by substituting those in the general series, Van-+ Van=2b+ Van-312+ &c.; it will become, 33+ 3/32+ 329=39+ 3/6+ x4, which is therefore the multiplier ; whence '9+ 6+ 34 2
2(3/9+36+94)=2(3/9+3/6+ 34) 39+ 36+ 34*33-32
to a fraction which shall have a rational deno.
3x + y minator.
Here n=3, a=r, and b=y; therefore by substituting 3, x and y, for n, a and b in the general series, var--Var 36+ Varsfe+&c. gives 3.x - try+ 3y for the multiplier ; whence 32-3y + 3y
clar Yxy + y2)
x+y 5 5. Reduce 17+ 43
to a fraction which shall have a rational denomina.or.
Here n=4, a=7, and b=3 ; whence the series vamia vas+ Van-302-an-+63 will beconie $78-4723+ 4/7:39-43S which is the multiplier ; whence 4/7 - 272:3+ 47:3°— 4139 5
x 478-4723+27:32 – 43^27+ 43 5( V 343 — 4147 + 4/63 — 427)
4 On extracting the Roots of Binomial Surds. (140.) In some cases the square root of a binomial, one of whose terms is a quadratic surd, and the other a rational quantity, may be expressed by another binomial, one or both of whose terms are quadratic surds.
In order to determine a rule to effect this when it can be done ; let a+v6 be the given binomial, and let its root be rts. Then because watVb)=r+s, by squaring each side of this equation we obtain a t vb=rl + 2rs+s?. Now and s being either one or both quadratic surds, pl and so will be rational, and 2rs irrational : let go? + so=a and 2rs= vb; then squaring each side of the last two equations, and subtracting the corresponding sides of the latter from the former, we obtain pe-2r?go+s4=a'-6: but go — 2roge +5*, is the square of me — 52; whence (78—s?)=a?–6: therefore taking the roots of both sides ge-g=v(a-6): we have now the difference of qs and se expressed by the given quantities a and b; but since by assuniption, y +sa=a we have also their sum expressed in terms of the same given quantities.
Hence we have the sun and difference of two quantities given, to find the quantities themselves,
viz. po? +=a
and p-s=a-) now by adding these two equations we obtain
2ro=a+ v(a? — b) whence pe=a+v(2®—) and by subtracting the lower from the upper of the same two equations we get
a- v(a®-6) 2sora-v(a?-1) whence ge=
2 therefore extracting the roots of the sides of the last two equations, tv (amb)
v (al-b) we obtain r=N
2 The square root of the binomial surd a + vb can therefore only be exhibited under the form now shown when amb is a square number. In the same manner it may be shown, that the square root of
THE METHOD OF INDETERMINATE COEFFICIENTS.
Therefore to extract the root of a binomial or residual surd we must substitute the numbers or quantities of which the given surd is composed, in place of the letters in one of the formula, denoted by (A) and (B) acccording as the terms are connected by + or
Examples 1. Extract the square root of 11+w72, whence a=ll, and b=72 consequently
therefore ✓(11+72)=3+ 2.
2. Find the square root of 3—272.
Ans. w5+1. 4. Find the square root of 23–827.
Ans. 4+7 5. Find the square root of 36+ 10/11. 6. Find the square root of 33+12v6.
On the method of Indeterminate Coefficients. (141.) The method of Indeterminate Coefficients, which is of the greatest utility in the higher branches of the mathematics, is particuJarly applicable to the resolution of the problem under consideration. It depends upon the following theorem.
THEOREM. Let r denote an indeterminate quantity, that is a quantity which may have any value whatever, and let A, B, C, &c. and A, B, C, &c. be quantities which are entirely independent of I, then if the two expressions
A + BI + Cı? + Dx",..
A' + B'r + C'x? + D'rs.. which may be supposed continued to any number of terms, be equal to one another, the coefficients of the like powers of c in both must be equal, that is, A=A', B=B', C=C, &c.