Elements of Algebra

CHAP. XVII. OF ALGEBRA. 53 $a^{-10}$ multiplied by $a$, gives $a^{-9}$, and so on. [See Art. 175, 176.] 182. Next, if it be required to multiply any power of $a$ by $a^2$, or the second power, I say that the exponent becomes greater by $2$. Thus, the product of $a^3$ by $a^2$ is $a^5$; that of $a^4$ by $a^2$ is $a^6$; and more generally, $a^n$ multiplied by $a^2$ makes $a^{n+2}$. With regard to negative exponents, we shall have $a^1$, or $a$, for the product of $a^{-1}$ by $a^2$; for $a^{-1}$ being equal to $\frac{1}{a}$, it is the same as if we had divided $aa$ by $a$; consequently, the product required is $\frac{aa}{a}$, or $a$; also $a^{-2}$, multiplied by $a^2$, produces $a^0$, or $1$; and $a^{-3}$, multiplied by $a^2$, produces $a^{-1}$. 183. It is no less evident, that to multiply any power of $a$ by $a^3$, we must increase its exponent by three units; consequently, the product of $a^n$ by $a^3$ is $a^{n+3}$. And whenever it is required to multiply together two powers of $a$, the product will be also a power of $a$, and such that its exponent will be the sum of those of the two given powers. For example, $a^5$ multiplied by $a^8$ will make $a^{13}$, and $a^{12}$ multiplied by $a^7$ will produce $a^{19}$, \ldots 184. From these considerations we may easily determine the highest powers. To find, for instance, the twenty-fourth power of $2$, I multiply the twelfth power by the twelfth power, because $2^{24}$ is equal to $2^{12} \times 2^{12}$. Now, we have already seen [Table, p. 49] that $2^{12}$ is $4096$; I say therefore that the number $16777216$, or the product of $4096$ by $4096$, expresses the power required, namely, $2^{24}$. 185. Let us now proceed to division. We shall remark, in the first place, that to divide a power of $a$ by $a$, we must subtract $1$ from the exponent, or diminish it by unity; thus, $a^5$ divided by $a$ gives $a^4$; and $a^0$, or $1$, divided by $a$, is equal to $a^{-1}$ or $\frac{1}{a}$; also $a^{-3}$ divided by $a$, gives $a^{-4}$. 186. If we have to divide a given power of $a$ by $a^2$, we must diminish the exponent by $2$; and if by $a^3$, we must subtract $3$ units from the exponent of the power proposed; and, in general, whatever power of $a$ it is required to divide by any other power of $a$, the rule is always to subtract the exponent of the second from the exponent of the first of those powers: thus $a^{15}$ divided by $a^7$ will give $a^8$; $a^6$ divided by $a^7$ will give $a^{-1}$; and $a^{-3}$ divided by $a^4$ will give $a^{-7}$. 187. From what has been said, it is easy to understand