Page:Elementary algebra (1896).djvu/187

CHAPTER XX.

Involution.

185. Definition. Involution is the general name for repeating an expression as a factor, so as to find its second, third, fourth, or any other power. Involution may always be effected by actual multiplication. Here, however, we shall give some rules for writing at once

(1) any power of a monomial; (2) the square and cube of any binomial; (3) the square and cube of any multinomial; (4) any power of a binomial expressed by a positive integer.

186. It is evident from the Rule of Signs that

(1) no even power of any quantity can be negative; (2) any odd power of a quantity will have the same sign as the quantity itself.

Note. It is especially worthy of notice that the square of every expression, whether positive or negative, is positive.

INVOLUTION OF MONOMIALS.

187. From definition we have, by the rules of multiplication,

$\left(a^{2}\right)^{3}$	$=$	$a^{2}\cdot a^{2}\cdot a^{2}=a^{6}$.
$\left(-x^{3}\right)^{2}$	$=$	$\left(-x^{3}\right)\left(-x^{3}\right)=x^{3+3}=x^{6}$.
$\left(-a^{5}\right)^{3}$	$=$	$\left(-a^{5}\right)\left(-a^{5}\right)\left(-a^{5}\right)=-a^{5+5+5}=-a^{15}$.
$\left(-3a^{3}\right)^{4}$	$=$	$\left(-3\right)^{4}\left(a^{3}\right)^{4}=81a^{12}$.

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