taking care to give them their proper exponents. 2 am2 × 3cd is the same as 2 amm × 3ccdd, which gives 6ammccdd6 am2 c2 d2. a3 multiplied by a2 gives a3 a2; but a3 = a a a and a2 — a a ; hence a3 a2 = aa a aa = a3. In all cases the product consists of all the factors of the multiplicand and multiplier. In the last example a is three times a factor in the one quantity, and twice in the other; hence it will be five times a factor in the product. The exponents show how many times a letter is a factor in any quantity; hence if any letter is contained as a factor one or more times in both multiplier and multiplicand, the exponents being added together will give the exponent of that letter in the product. ax a = a1× a1 a11 = a2. a2 × a' = a2+1 = a3. It has already been remarked that the addition of two or more quantities is performed by writing the quantities after each other with the sign+between them. The sum of 3 ab, 2 a cd, 5 a2 b, 4 a b, and 3a2 b, is 3 a b + 2 a c d + 5 a2 b + 4 ab +3 ab. But a reduction may be made in this expression, for 3ab4ab is the same as 7 ab; and 5 a2 b + 3 a b is the same as 8 a3b; hence the expression becomes 7ab2acd+8 a2 b. Reductions of this kind may always be made when two or more of the terms are similar. When two or more terms are composed of the same letters, the letters being severally of the same powers, they are said to be similar. The numerical coefficients are not regarded. The quantities 4 a b and 3 a b are similar, and so are 5 a2 b and 3 a2 b; but 4 a b and 5 a2 b are not similar quantities, and cannot be united. The subtraction of algebraic quantities is performed by writing those, which are to be subtracted, after those from which they are to be taken, with the sign between them. 5 ab2 to be This last ex Ifb is to be subtracted from a it is written a b. subtracted from 8 a b2, is written 8 a b2. -5 a b2. pression may be reduced to 3 ab. In all cases when the quantities are similar, the subtraction may be performed immediately upon the coefficients. Compound Quantities. XI. The addition and subtraction of simple quantities, produce quantities consisting of two or more terms which are called compound quantities. 2a + c d -3b is a compound quantity. Addition of Compound Quantities. The addition of two or more compound quantities, when all the terms are affected with the sign + will evidently be the same, as if it were required to add together all the simple quantities of which they are composed; that is, they must be written one after the other with the sign + before all the terms except the first. The sum of the quantities 3 a + 2 c and b + 2d is 3 a + 2 c + b + 2 d. If the quantities 3 a b + 5d and b c be added, in which some of the terms have the sign the sum will be 3 a b + 5 d +b-c; for b c is less than b, therefore, ifb be added the sum will be too large by the quantity c. Hence c must be subtracted from the result. This may be illustrated by figures. Add together 17+ 10 and 20 6. Now 20 6 is 14 and 17+ 10 + 20 — 6 is equal to 17+ 10 + 14. From the above observations we derive the following rule for the addition of compound quantities. Write the quantities after each other without changing their signs, observing that terms which have no sign before them are understood to have the sign +. A sign affects no term except the one immediately before which it is placed; hence it is unimportant in what order the terms. are written, for 14 5+2 has the same value as 14+ 5+2+14. Those which have the sign + 2-5 or as are to be added together, and those which have the sign — are to be subtracted from their sum. If the first term has the sign +, the sign may be omitted before this term, but the sign must always be expressed. Great care is requisite in the use of the signs, for an error in the sign makes an error in the result of twice the quantity before which it is written. Add together 3a+2bc-3 c and 4 5a-3bc2+2 c 3a+2bc2-3c+5a-3bc2+2c+7 ab But this expression may be reduced. and 3a5aa8aa7a, 2bc-3bc4 b c — 2 b c2 = 6 b c2 — 5 b c2 = b c2, and -3c2c8c+3c=-11c+5c-6c; hence the above quantity becomes 7 a + b c2 + 7 ab-6 c1. To reduce an algebraic expression to the least number of terms, collect together all the similar terms affected with the sign + and also those affected with the sign, and add the coefficients of each separately; take the difference of the two sums and put it into the general result, giving it the sign of the larger quantity. 3. Add together the following quantities. 7 mab― 16— 43 m y, and and and and xy-ax-ay+axy, nb. -2xy-2ay + 3a x + 15, -15 a xy-13 am +43 + 18 arx, 5. Add together the following quantities. XII. The subtraction of simple quantities, as has already been observed, is performed by giving the sign to the quantity to be subtracted, and writing it before or after the quantity, from which it is to be taken. If it is required to subtract c+d from a + b it is plain that the result will be a + b C d, for the compound quantity c + d is made up of the simple quantities c and d, which being subtracted separately would give the above result. From 22 subtract 13-7. and 22 13-76. 22-616. The result then must be 16. But to perform the operation on the numbers as they stand, first subtract 13, which gives 139. This is too small by 7 because the number 13 is larger by 7 than the number to be subtracted, therefore in order to obtain a correct result the 7 must be added; thus 22 13716, as required. First subtract b, which gives a-b. This quantity is too small by c because 6 is larger than be by the quantity c. Hence to obtain a correct result c must be added, thus a ·b+c. This reasoning will apply to all cases, for the terms affected with the sign-in the quantity to be subtracted diminish that quantity; hence if all the terms affected with + be subtracted, the result will be too small by the quantities affected with -, these quantities must therefore be added. The reductions may be made in the result, in the same manner as in addition. Hence the general RULE. Change all the signs in the number to be subtracted, the signs to, and the signs + -to, and then proceed as in addition. ·2 a2x+2by-3 ac3 +6 3b x2-7a x2+13 13bc-3 ax3 — 8. Ans. 3b x2-13bc-4 a x3 + 21. 17 a2y + 13 ay—a—3 2ay-b-11a +5. 42 axy-4 a x 143-17 y ax3 |