Elements of Algebra

52 ELEMENTS SECT. I 179. Lastly, we have to consider the powers of negative numbers. Suppose the given number to be $-a$; then its powers will form the following series: $-a, +a^2, -a^3, +a^4, -a^5, +a^6$, &c. Where we may observe, that those powers only become negative, whose exponents are odd numbers, and that, on the contrary, all the powers, which have an even number for the exponent, are positive. So that the third, fifth, seventh, ninth, &c. powers have all the sign $-$; and the second, fourth, sixth, eighth, &c. powers are affected by the sign $+$. CHAP. XVII. Of the Calculation of Powers. 180. We have nothing particular to observe with regard to the Addition and Subtraction of powers; for we only represent those operations by means of the signs $+$ and $-$, when the powers are different. For example, $a^3 + a^2$ is the sum of the second and third powers of $a$; and $a^5 - a^4$ is what remains when we subtract the fourth power of $a$ from the fifth; and neither of these results can be abridged: but when we have powers of the same kind or degree, it is evidently unnecessary to connect them by signs; as $a^3 + a^3$ becomes $2a^3$, &c. 181. But in the Multiplication of powers, several circumstances require attention. First, when it is required to multiply any power of $a$ by $a$, we obtain the succeeding power; that is to say, the power whose exponent is greater by an unit. Thus, $a^2$, multiplied by $a$, produces $a^3$; and $a^3$, multiplied by $a$, produces $a^4$. In the same manner, when it is required to multiply by $a$ the powers of any number represented by $a$, having negative exponents, we have only to add $1$ to the exponent. Thus, $a^{-1}$ multiplied by $a$ produces $a^0$, or $1$; which is made more evident by considering that $a^{-1}$ is equal to $\frac{1}{a}$, and that the product of $\frac{1}{a}$ by $a$ being $\frac{a}{a}$, it is consequently equal to $1$; likewise $a^{-2}$ multiplied by $a$, produces $a^{-1}$, or $\frac{1}{a}$; and