Elements of Algebra

CHAP. XVI. OF ALGEBRA. 49 power, because a certain given number has been multiplied by itself; and if a number has been multiplied twice by itself we call the product the third power, which therefore means the same as the cube; also if we multiply a number three times by itself we obtain its fourth power, or what is commonly called the *biquadrate*: and thus it will be easy to understand what is meant by the fifth, sixth, seventh, &c. power of a number. I shall only add, that powers, after the fourth degree, cease to have any other but these numeral distinctions. 170. To illustrate this still better, we may observe, in the first place, that the powers of $1$ remain always the same; because, whatever number of times we multiply $1$ by itself, the product is found to be always $1$. We shall therefore begin by representing the powers of $2$ and of $3$, which succeed each other as in the following order: Powers. Of the number 2. Of the number 3. 1st $2$ $3$ 2d $4$ $9$ 3d $8$ $27$ 4th $16$ $81$ 5th $32$ $243$ 6th $64$ $729$ 7th $128$ $2187$ 8th $256$ $6561$ 9th $512$ $19683$ 10th $1024$ $59049$ 11th $2048$ $177147$ 12th $4096$ $531441$ 13th $8192$ $1594323$ 14th $16384$ $4782969$ 15th $32768$ $14348907$ 16th $65536$ $43046721$ 17th $131072$ $129140163$ 18th $262144$ $387420489$ But the powers of the number $10$ are the most remarkable: for on these powers the system of our arithmetic is founded. A few of them ranged in order, and beginning with the first power, are as follow: 1st 2d 3d 4th 5th 6th $10$, $100$, $1000$, $10000$, $100000$, $1000000$, &c. 171. In order to illustrate this subject, and to consider it in a more general manner, we may observe, that the E