When the denominators of algebraic fractions differ it is necessary before addition or subtraction can be performed to harmonize them, as in the reduction of the denominators of numerical fractions (Arts. 388-390). For example, add together the fractions a/bc e/b, r/ac In these denominators we perceive that they collectively contain the letters a, b and c, and no others. It will be requisite, therefore, that each of the fractions be modified so that its denominator shall have these three factors. To effect this it will be seen that it is necessary to multiply each fraction by that one of these letters which is lacking in its denominator. Thus, in the first, a is lacking, therefore (Art. 380) a x a = aa In the second a
bc x a = abc
and c are lacking, therefore e x ac = ace and in the third
b x ac = abc,
b is lacking, therefore r xb = rb Placing them now
acxb = abc
together we have -
aa+ace+br/abc = a/bc+e/b + r/ac
The factor a a may be represented thus a2, which means that a occurs twice, the small figure at the top indicating the number of times the letter occurs; a2 is called a squared, a a a = a3, and is called a cubed.
In order to show that the above fraction, resulting as the sum of the three given fractions, is correct, let a = 2, b = 3, c = 4, e = 5, and r - 6. Then the three given fractions are -
2/3x4+5/3+6/2x4 = 1/6+5/3+3/4.
In equalizing these denominators we multiply the second fraction by 2, and the third by 1 1/2, which will give -
5 x 2 = 10, 3 x 1 1/2 = 4 1/2;
3 2 6 4 1 1/2 6
1 + 10 + 4 1/2 = 15 1/2 = 2 3 1/2 = 2 7
6 6 6 6 6 12
Now the sum of the fractions is -
4 + 40+ 18 62 14 = 7 ;
or, = = 2 2
24 24 24 12
the same result as before, thus showing that the reduction was rightly made.