= e a f- bdr با ac Fractions are changed to others of equal value Ex + Here with a common denominator, by multiplying each f numerator by every denominator ercept its own, a is considered as a fraction whose denoinifor the new numerator ; and all the denomina nator is unity. turs together for the common denominator, If two fractions have a common denominator, their difference is found by taking the difference Let i'd's be the proposed fractions; then of the numerators and retaining the common deadich redh nominator. Thus, are fractions of the same bdf'odi'if' value with the former, having the common 6 cbf denominator bdf. For If they have not a common denominator, edb and the numerator and deno they must be transformed to others of the j à Tdji same value, which have a common denomiminator of each fraction having been multi nator, and then the subtraction may take plied by the same quantity, viz. the product place as above. of the denominators of all the other fractions. When the denominators of the proposed hr ad - hc Ex. 2. fractions are not prime to each other, find 6 bd bd b d their greatest common measure; multiply both the numerator and denominator of each al ah -od fraction, by the denominators of all the rest, Ex. 3. a-divided respectively by their greatest coinmon measure; and the fractions will be re c+d ac-ad bctbd duced to a common denominator in lower Ex. 4. 7 bc-od 6c-one terms than they would have been by proceed adhc-hd ing according to the former rule. bc-bd 6 Thus, reduced to a The sign of bd is negative, because every m r' my' m 2 part of the latter fraction is to be taken from bra mon denominator, are the former. m.xyz m x y z сту ON TIIE MULTIPLICATION AND DIVISION OF in y y z FRACTIONS. To multiply a fraction by any quantity, mulON THE ADDITION AND SUBTRACTION OF tiply the numerator by that quantity and retain FRACTIONS. the denominator. If the fractions to be added have a common Thus, хс — 7 For if the quantity to 6 denominator, their sum is found by adding the numerators together and retaining the common be divided be c times as great as before, and denominator. Thus, the divisor the same, the quotient must be c times as great. ato The product of two fractions is found by multiplying the numerators together or a new If the fractions have not a common deno numerator, and the denominators for a new deminator they must be transformed to others nominator. of the same value, which have a common de Let and be the two fractions; then sominator, and then the addition may take 6 d place as before. х =y, by ad adtbc Ex. 2. + it Ē b a+b Ex. 3. therefore, ac=bd xy; dividing these equal quantities, ac and bdxy, by bd, we have -htatb =ry bd a com a ас = с For if a a + a Ωα a bf-; and ; the ас ас C - ab X =t, and To divide a fraction by any quantity, multi to the denominator, and the contrary, by ply the denominator by that quantity, and relain changing the sign of its index. Thus, the numerator. a" x 9" 0" x 9" be a" b? bp The fraction divided by c, isão Because ON INVOLUTION AND EVOLUTION. and a cth part of this is INVOLUTION. If a quantity be continually ūbe bc multiplied by itself, it is said to be involved, quantity to be divided being a cth part of or raised; and the power to which it is raiswhat it was before, and the divisor the saine. ed is expressed by the number of times the The result is the same, whether the deno- quantity has been employed in the multipliminator is multiplied by the quantity, or the cation. numerator divided by it. Thus, a X a, or a?, is called the second Let the fraction be if the denominator power of a; a Xa xa, or a', the third power; a Xa.... (n), or an, the nth power. be multiplied by c, it becomes or õds the signs of the even powers will be positive, , bic the quantity which arises from the division and the signs of the odd power negative. of the numerator by c. For -a X -a= a?; -ax-ax -a, &c. To divide one fraction by another, invert the A simple quantity is raised to any power, numerator and denominator of the divisor, and by multiplying the index of every factor in proceed as in multiplication. the quantity by the exponent of the power, and prefixing the proper sign determined by Let į and à à the last article. Thus, a" raised to the nth power is ann. BeХ āc' causer" xam xa" ... to n factors, by the rule of multiplication, is a "."; also, all'= a For if =y, then a =bx, ab xab x &c. to n factors, or a Xa xa to n factors x bxbxb.,.. to na and c=dy; also, ad = bdx, and bc= ad bd x factors = an X 6"; and a2 b3 c raised to the bdy; therefore fifth power is aloh15 c. Also, -q" raised to bc the nth power is #amn; where the positive or The rule for multiplying the powers of the negative sign is to be prefixed, according as same quantity will hold when one or both n is an even or odd number. of the indices are negative. If the quantity to be involved be a fracThus, an xa-=a~; for a" x 9- tion, both the numerator and denominator a" x =9*-*; must be raised to the proposed power. If the quantity proposed be a compound one, the involutiou may either be represent ed by the proper index, or it may actually Again, a-m xa = a-nt; because a-m take place. Let a to be the quantity to be raised to x = any power. If m = n, a' = = a'; also, at6 a" X 4-7 = = 1; therefore a =1; a2 tab according to the notation adopted. tab + b2 The rule for dividing any power of a quan a xoor al + 2ab + b2 the sq. or 24 power tity by any other power of the same quan a + b tity holds, whether those powers are positive a3 + 2ab + ab? or negative. + ab + 2 ab2 + 63 Thus, am an X a" = a toor aš +3226+ 3 a 62 +63 the 3d p. a +b 1 at + 3aib + 3a:62+ a 63 Again, a---2-= • + a3b + 3 9262 + 3 abstbe Hence it appears, that a quantity may be a to?* or a+ + a36 + 6 a:b: + 4 abs tbe transferred from the numerator of a fraction the fourth power. Q 1 1 1 arta If b be negative, or the quantity to be in- a volved be ab, wherever an odd power of or a? x 6-1, and or ai x 6? b enters, the sign of the term must be nega. tive. To extract the square root of a compound Hence, a D]* = a4 - 4 a36+6 abquantity. - 4 ab3 + 64. a+ 2ab + b2(a + b EVOLUTION, or the extraction of roots, is the method of determining a quantity which 2a + b)2 ab +62 raised to a proposed power will produce a 2 ab + b2 given quantity. Since the nth power of am is am", the nth Since the square root of ar + 2ab + b2 is root of qmn must be a*; i. e. to extract any at 6 whatever be the values of a and b, we root of a single quantity, we must divide the may obtain a general rule for the extraction index of that quantity by the index of the of the square root, by observing in what root required. manner a and b may be derived from ast When the index of the quantity is not 2 ab +62 exactly divisible by the number which ex Having arranged the terms according to presses the root to be extracted, that root the dimensions of one letter, a, the square must be represented according to the nota root of the first term, a?, is a, the first factor tion already pointed out. in the root; subtract its square from the Thus, the square, cube, fourth, nth root of whole quantity, and bring down the remainai + x?, are respectively represented by der 2 ab + be; divide 2 ab by 2 a, and the q3+, w7x), a2 +3zt, aztxat; result is b, the other factor in the root; then 1 multiply the sum of twice the first factor and the same roots of or a? + x1-', are the second (2a + b), by the second (6), and subtract this product (2 ab + b) from represented bya?ti=1-1992+x=1-4,0407, the remainder. If there be no more terms, consider a + b as a new value of a; and its If the root to be extracted be expressed square, that is az +206 + 62, having, by by an odd number, the sign of the root will the first part of the process, been subtracted be the same with the sign of the proposed from the proposed quantity, divide the re mainder by the double of this new value of a, quantity. If the root to be extracted be expressed by subtrahend, multiply this factor hy twice the for a new factor in the root; and for a new an even number, and the quantity proposed sum of the former factors increased by this be positive, the root may be either positive or factor. The process must be repeated till negative. Because either a positive or nega- tive quantity, raised to such a power, is the root, or the necessary approximation to the root, is obtained. positive. If the root proposed to be extracted be Ex. 1. To extract the square root of a? + expressed by an even number, and the sign 2ab + b2 + lac +2bctc2. of the proposed quantity be negative, the ai + 2ab+ba+2ac+2bc+c(a+b+c root cannot be extracted; because no quan- a2 tity, raised to an even power, can produce 2a + b)2 ab + b2 a negative result. Such roots are called 2 ab + b2 impossible. 2a +26+0)* Any ioot of a product may be found by 2ac+2bctc? taking that root of each factor, and multi fact?betc? plying the roots, so taken, together. =an x ba; because each of Ex. 2. To extract the square root of a? these quantities, raised to the nth power, is + ab. a? ar + In a=b, then ai xam uñ; and in the Thus, a bhatt 1 2 a2 1 +*(1+-&c. 1 + 10. 2+:-) + 8 square, or a xing a' 5 r Ex. 3. To extract the square root of 1+1. quantity be n, it is called an equation of * dimensions. In any equation, quantities may be transposed from one side to the other, if their signs be changed, and the two sides will still be equal. Let r + 10 = 15, then by subtracting 10 from each side, 5+10 — 10 = 15 — 10, or = 15 Let x 4 = 6, by adding 4 to each =y; adding a - -b to each 64 side, I -- a tbta-b=ytu-b; or yta - b. &c. Hence, if the signs of all the terms on 64 each side be changed, the two sides will still It appears from the second example, that be equal. Let x - a=b- 2 .r; by transposition, a trimonial al -ext, in which four -6+2x=-rta; or a—=2x-b. times the product of the first and last terms If every term, on each side, be multiplied by is equal to the square of the middle term, is the same quantity, the results will be equal. 2 a complete square, or ax x 4 = a?r? An equation may be cleared of fractions, by multiplying every term, successively, by The method of extracting the cube root is the denoininators of those fractions, exceptdiscovered in the same manner. ing those terms in which the denominatons ai + 3a2b + 3 a 62 +63(a + b are found. Let 3 x + 3 a?) = 34; multiplying by 4, 3a2b + 3 a b2 +63 Sa? b + 3 a b2 +63 12r +5r=136, or 17x=136. The cube root of a'+322 6+3 ab'+b'is a If each side of an equation be divided by the +b; and to obtain a + b from this com same quuntity, the results will be equal. pound quantity, arrange the terms as before, 136 Let 17 x = 136; then r = and the cube root of the first term, a', is a, 17 the first factor in the root; subtract its cube If each side of an equation be raised to the from the whole quantity, and divide the first . same power, the results will be equal. term of the remainder by 3 as, the result is b, the second factor in the root; then subtract Let x1 = 9; then r= 9 X 9=81. 3a36+3 ab? + b3 from the remainder, and Also, if the same root be extracted on both the whole cube of a +b has been subtracted. sides, the results will be equal. If any quantity be left, proceed with a + b Let x=81; then xi = 9. as a new a, and divide the last remainder by 3.a 767' for a third factor in the root; and To find the value of an unknown quantity in thus any number of factors may be obtained. a simple equation. Let the equation first be cleared of fracON SIMPLE EQUATIONS. tions, then transpose all the terms which inIf one quantity be equal to another, or to volve the unknown quantity to one side of nothing, and this equality be expressed alge- the equation, and the known quantities to braically, it constitutes an equation. Thus, the other ; divide both sides by the co-effi. -Q=b - 3 is an equation, of which x cient, or sum of the co-efficients, of the an- a forms one side, and b - the other. known quantity, and the value required is When an equation is cleared of fractions obtained. and surds, if it contain the first power only of an unknown quantity, it is called a simple Ex. 1. To find the value of x in the equaequation, or an equation of one dimension: if tion 3 r — 5 = 23 the square of the unknown quantity be in any by transp. 3.x + x = 23 + 5 term, it is called a quadratic, or an equation or 4 r = 28 28 of two dimensions; and in general, if the loy division index of the highest power of the unknown = 8. 7. 2r or } de-cy — са =denty -ba ca de ba 11 C6 Cb са — Ex. 1. Let 3x - 5y = 13 To find .r and y. 24 Method. Find an expression for one of Ex. 2. Let .r+ +-=4r – 17. the unknown quantities, in each equation; Mult. by 2, and 2 r tr = 8x 34 put these expressions equal to each other, and 3 from the resulting equation the other unMult. by 3, and 61 +3.r - 2x=248-102 known quantity may be found. by transp. 6x +3r 2r - 24.7 --- 102 17.1= 102 Let To find r and y. 17 r = 102 Lbx toy=des From the first equat. x =Y from the second, b.x =de-cy, and r = b ba de-cy 1+ therefore ay = b I tha=car ba - by=de - су I car=-bd cy - by=de- b.y=de-ba de bn C-6 1 Also, r= y; that is, ca-ba---det be - de 3d Method. If either of the unknown quan12 tities have the same co-efficient in both equa3x 5 2 r - 4 tions, it may be exterminated by subtracting, Ex. 5. x+ 2 3 or adding, the equations, according as the 4 x 8 sign of the unknown quantity, in the two 2 x + 3r -5= 24 3 cases, is the same or different. Let To find r and y. 19 r = 95 95 By subtraction, 2 y = 8, and y = 4 By addition, 2 x = 22, and x = 11. If there be two independent simple eqna If the co-efficients of the unknown quantity tions involving two unknown quantities, they to be exterminated be different, multiply the may be reduced to ove which involves only terms of the first equation by the co-efficient one of the unknown quantities, by any of the of the unknown quantity in the second, and following methods: the terms of the second equation by the co10 Method. In either equation, find the efficient of the same unknown quantity in value of one of the unknown quantities in the first ; then add, or subtract, the resultterms of the other and known quantities, and ing equations, as in the former case. for it substitute this value in the other equation, which will then only contain one un ş known quantity, whose value may be found 22x+7 y by the rules before laid down. Multiply the terms of the first equation by 2, x+y=10 and the terms of the other by 3, Let To find x and 72.0-Sy=59 Y. then 6 r — - 10y = 26 From the first equat. x = 10 -y; hence, 2.x 6 x + 21 y = 243 = 20 — 2 y, By subtraction, 31 y=-217 by subst. 20 - 2y — 3y = 5 217 =7; 20 — 5 = 2y +3y 31 15 = 5 y also, 3.x - 5y = 13, or 5 x — 35 = 15 15 therefore, 3x = 13 + 35 = 48 5 48 and r =* = 16. hence also, r = 10-y=10-3=7. 3 = 12 -y=7 55. =81 |