Note. If the surd part is not the same in the quantities which are to be added or subtracted from each other, it is evident that the addition or subtraction can only be performed by placing the signs + or — between 9. Required the sum of 2/48 and 93108.... Ans. 8 √3 +27 3/4. (134.) The Multiplication and Division of Surd quantities. Rule.-Reduce the quantities to equivalent ones with the same index, and then multiply or divide both the rational and the irrational parts by each other respectively. =15x2x/10=30/10. 3. Multiply 24/3 by 34. By reduction, 2/3=2×38=2× √3-2√27, 3 6' and 33 4=3×43=3 /4 3/16 Hence 2√3×334=3√27 × 3√166√432 4. Divide 2 bc by 3 ac 5. Divide 103 108 by 54. a3c Now 10108=1027 x 4=13x3 x 4-304; 10108 304-6; or 10.3108 534 5 34 534 2272x3=6 Ans. $225000 Ans. 15. (135.) On the Involution and Evolution of Surd quantities. Rule.-Raise the rational part to the power or root required, and then multiply the fractional index of the surd part by the index of that power or root. (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 a both the numerator and denominator of the fraction be multiplied by √, and it becomes /; and by multiplying the numerator and denominator of the fraction comes b(a+x)3_b(a+x)3. Or in general, if both the numerator (a+x)s a+x a and denominator of a fraction of the form be multiplied by √2a1, 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, +x^==2y+x=-3y3+ &c...+y"-1 to n terms, whe ther n be even or odd. 3. xr+y" ·x"—2y + x^—3y2 —&c.. —y"-1 to n terms, when ·=x"—1 — x”—2y +x′′¬3y2—&c..+y" to n terms, when n is an odd number.* (138.) Now let "=a, y"=b, then x="√/a, y=√b, and these and a-b a-b a+b fractions severally become and Va−b' a+b Na+ Nii by the application of the rules, already laid down, we have x^~1="~√a^~1; x^~^=~/a^~2; x^—3="~√a^-3, &c.; also, y2= "[/b2; y3="√/b3, &c. hence, x"2y = √a^~2 × √b = "a"-2b; x#—3y2 = n n √a^-3x "~/b2="√a^-362, &c. By substituting these values of "-", x"-1y, x-3y2, &c. in the several quotients we have a-b Va-b n ;= ~an~+ Wan-21 + √a"−3b2 + &c.. + ~/b" to n terms; where the terms band whole number whatever. And / have the sign +, when n is an odd number; and the sign -> when n is an even number. n n (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 ab be multiplied by a+ ~a"¬3b+ √a"¬3b2+&c....+ √b (n being any whole number whatever), the product will be a-b, a rational quantity; and if a binomial surd of the form √a+√b be multiplied by a ~~—"/a"~1b+"√/a"-3b2—&c....±√/b′′-1, the product will be -y =x+y; xy4 x-y ·=x3+x2y+xy2+y3, &c. x2-y2 2. x+y x+y x+y x3+y3_x2—xy+y2; a+b or a−b, according as the index n is an odd or an even number; but here we must 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. Now a√x=√/a2 — √x ; put a2=c, then a2— √x=√c−√x. In this example n is equal to 2, and therefore the number of terms to be taken of the general series +ab+&c. is 2, and because a2=c and bx, the multiplier is c+ x; whence Here again n=2: therefore the number of the terms to be taken of the series, /a”—' — Wa”—2b+&c. is 2; and in this example, a=8 and b=3: therefore the multiplier is 8-√3; whence √8-√3 √6 43-32 √8-√3^ √8+ √3` X 5 2 3. Reduce minator. to a fraction which shall have a rational deno 33-32 n Here n=3, a=3, and b=2; therefore by substituting those in the general series, an~ + √an-2b+ √a-36+ &c.; it will become, 32+3·2+322=39+ 36+ 4, which is therefore the multiplier; whence 39+36+34 2 = 2(39+Y6+Y4)_2(¥9+Y6+Y4) 3-2 to a fraction which shall have a rational deno n n Here n=3, a=r, and b=y; therefore by substituting 3, x and y, for n, a and b in the general series, a√an-2b+ √an-3b2 + &c. x2-xy+y for the multiplier; whence Yx2—Y xy+Yy2 Zx2-Yxy+Yy2 Z x + Zy gives |