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It frequently happens, as in some of the above examples, that a quantity is multiplied several times by itself, or enters several times as a factor into a product; as 3 a a a b b, into which a enters three times and b twice as a factor. In cases like this the expression may be very much abridged by writing it thus, 3 a” b°. That is, by placing a figure a little above the letter, and a little to the right of it, to show how many times that letter is a factor in the product. The figure 3 over the a shows, that a enters three times as a factor; and the 2 over the b, that b enters twice as a factor, and the expression is to be understood the same as 3 a a a b b. The figure written over the letter in this manner is called the index or exponent of that letter. The exponent affects no letter except the one over which it is Written. Care must be taken not to confound exponents with coefficients. The quantities 3 a and a” have very different values. Suppose a = 4, then 3 a = 12; whereas a” = 4 × 4 × 4 = 64. In the product 3 a” b” suppose a = 4 and b = 5, then 3 a” b” = 3 × 4 × 4 × 4 × 5 × 5 = 4800. The expression a” is called the second power of a, a” is called the third power, at the fourth power, &c. To preserve a uniformity, a, without an exponent, is considered the same as a , which is called the first power of a.” Figures as well as letters may have exponents. The first power of 3 is written
The multiplication of quantities in which some of the factors are above the first power, is performed in the same manner as in other cases, by writing the letters of both quantities together,
*In most treatises on algebra a” is called the square of a, and as the cube of a. The terms square and cube were borrowed from geometry, but as they are not only inappropriate, but convey ideas very foreign to the present subject, it has been thought best to discard them entirely.
taking care to give them their proper exponents. 2 a m”X 3 co d” is the same as 2 a m m × 3 c c d d, which gives 6 a mm cc d d = 6 a m” co d”. a multiplied by d” gives a” a”; but a' = a a a and a' = a a, hence a” a” = a a a a a = a”. 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 expoments being added together will give the exponent of that letter in the product. a × a = a1 × a = a + = a”. a” X a = a”** = ao. as X a” = a +* = a *, &c.
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 & b, 2 a cd, 5 a” b, 4 a b, and 3 a” b, is 3 a b + 2 a c d – 5 a” b + 4 a b + 3 a” b. But a reduction may be made in this expression, for 3 a b + 4 a b is the same as 7 a b; and 5 a” b + 3 a” b is the same as 8 a” by hence the expression becomes
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 a.” b and 3 a” by but 4 a b and 5 a” 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.
If b is to be subtracted from a it is written a – b. 5 a bo to be subtracted from 8 a b”, is written 8 a b” — 5 a b”. This last expression may be reduced to 3 a b”. In all cases when the quantities are similar, the subtraction may be performed immediately upon the coefficients.
XI. The addition and subtraction of simple quantities, produce quantities consisting of two or more terms which are called compound quantities. 2 a + c d – 3 b 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 + 2 d is 3 a + 2 c + b + 2 d. If the quantities 3 a b + 5 d 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, if b 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–H 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 + 2 — 5 or as — 5 + 2 + 14. Those which have the sign + 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.
1. Add together the following quantities. 5 a b – 2 a” m
and 3 a b — 5 a m + 2 a m.
2. Add together the following quantities. 13 a n” — 6 m + 2*,
and 7 b m — 32°– 8 y,
and 4 an” + 5 a z*-4 y.
XII. The subtraction of simple quantities, as has alread
been observed, is performed by giving the sign — to the quanti
to be subtracted, and writing it before or after the quantity, rom 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.
13 — 7 = 6.
and 22 – 6 = 16.
The result then must be 16. But to perform the operation on the numbers as they stand, first subtract 13, which gives 22 – 13 = 9. 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 — 13 + 7 = 16, as required.