Page:Elementary algebra (1896).djvu/194

 CHAPTER XXI.

Evolution.

195. The root of any proposed expression is that quantity which being repeated as a factor the requisite number of times produces the given expression. (Art. 14.)

The operation of finding the root is called Evolution: it is the inverse of Involution.

196. By the Rule of Signs we see that (1) any even root of a positive quantity may be either positive or negative;

(2) no negative quantity can have an even root;

(3) every odd root of a quantity has the same sign as the quantity itself.

Note. It is especially worthy of notice that every positive quantity has two square roots equal in magnitude, but opposite in sign.

Ex. ^(9<^Qfi) = -Zaj-

In the present chapter, however, we shall confine our attention to the positive root.

EVOLUTION OF MONOMIALS.

197. From a consideration of the following examples we will be able to deduce a general rule for extracting any proposed root of a monomial.

Examples. (1) V(«*^)= «*^ because {(i^y= cfib*. (2) ^(- 2?) = - x» because (-ac»)» = -z®. (3) ^(<«>) = c* because (c*)» = c». (4) ^(81 x«) = 3 z5 because (3 x»)* = 81 x^. 176