An Elementary Treatise on Diff. Eqns. and Their Applns. by H. Piaggio

By H. Piaggio

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If numerator and denominator are divided by the same number, the value of the fraction is unaltered. This can be expressed algebraically as follows: Let g be any fraction. —, Then a ax n b b x n a a -f- m and b~b+m' where m and » are any numbers. 3<> Example!. —In practice there is no need to write out the powers in full as shown above. The rule for-the division of powers may be applied directly. Example 2. Simplify x As in Arithmetic, factors in either numerator can be cancelled with factors in either denominator.

M. of the denominators —viz. 60. —It is not possible to cancel any factors of 8 or 25 with factors of 60. This mistake is sometimes made by beginners, although the proceeding is contrary to the laws of Arithmetic. Only factors common to each term of the numerator can be cancelled, with factors of the denominator. Example 5. Simplify ~ - j^p. M. of the denominators. M. M. of a*b and ab*. This is a2b2, since both of them will divide into it exactly. M. of the denominators. y _ (* X 3b) — (y x 2a) 12a2b 18ab2 ~ 36a*6» — 3bx — lay 24.

In the example above the product (axaxaxaxa) is the 5th power of a, and the index of the product is the sum of the indices of the two factors. a 2 x a3 = a 2 + 3 - a5. The same reasoning may be applied to other cases and so ELEMENTARY OPERATIONS IN ALGEBRA 31 we may deduce the general rule for the multiplication of two powers of a number. When two powers of the same number are multiplied, the index of the product is the sum of the indices of the factors. Examples. (1) X* X X* = X* + t = Xs. (2) 2a 7 X a 3 = 2 a ' + 3 = 2a 10 .

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