Page:Elementary algebra (1896).djvu/57

39 Rh DIVISION OF SIMPLE EXPRESSIONS.

56. The method is shown in the following examples:

Ex. 1. Since the product of 4 and x is 4x, it follows that when 4x is divided by x the quotient is 4, or otherwise, 4x \div x=4.

Ex. 2. Divide 27 a^5 by 9 a^3.

The quotient {27a^5}{ 9a^3}={27 aaaaa }{9aaa} = 3aa = 3a^2

Therefore 27a^5 \div 9a^3=3a^2.

Ex. 3. Divide 35 a^3b^2c^3 by 7 ab^2c^3.

The quotient ={ 35 aaa. bb.ccc}{7a.bb.cc}= 5 aa. c = 5a^2c

We remove from the divisor and dividend the factors common to both, as in Arithmetic.

We see, in each case, that the index of any letter in the quotient is the difference of the indices of that letter in the dividend and divisor. This is called the Index Law for Division.

We can now state the complete rule:

Rule. The index of each letter in the quotient is obtained by subtracting the index of that letter in the divisor froin that in the dividend.

To the result so obtained prefix with its proper sign the quotient of the coefficient of the dividend by that of the divisor.

Ex. 4. Divide 45 a^b^x^ by — 9a^bx^.

The quotient =(— 5) x a®-3U2-ly!-2 =— 6 @bz?.

Ex. §. — 21076? +(— 7a*b?)=38b.-

Nore. If we apply the rule to divide any power of a letter by the same power of the letter, we are led to a curious conclusion.

Thus, by the rule @=C=E2=0; 3 but also @+@=“ =1. e oma? = 1p

This result will appear somewhat strange to the beginner, but its full significance will be explained in the chapter on the Theory of Indices.