Elements of Algebra

54 ELEMENTS SECT. I. the method of finding the powers of powers, this being done by multiplication. When we seek, for example, the square, or the second power of $a^3$, we find $a^6$; and in the same manner we find $a^{12}$ for the third power, or the cube, of $a^4$. To obtain the square of a power, we have only to double its exponent; for its cube, we must triple the exponent; and so on. Thus, the square of $a^n$ is $a^{2n}$; the cube of $a^n$ is $a^{3n}$; the seventh power of $a^n$ is $a^{7n}$, &c. 188. The square of $a^2$, or the square of the square of $a$, being $a^4$, we see why the fourth power is called the *biquadrate*: also, the square of $a^3$ being $a^6$, the sixth power has received the name of the *square-cubed*. Lastly, the cube of $a^3$ being $a^9$, we call the ninth power the *cubo-cube*: after this, no other denominations of this kind have been introduced for powers; and, indeed, the two last are very little used. CHAP. XVIII. *Of Roots, with relation to Powers in general.* 189. Since the square root of a given number is a number, whose square is equal to that given number; and since the cube root of a given number is a number, whose cube is equal to that given number; it follows that any number whatever being given, we may always suppose such roots of it, that the fourth, or the fifth, or any other power of them, respectively, may be equal to the given number. To distinguish these different kinds of roots better, we shall call the square root, the *second root*; and the cube root, the *third root*; because, according to this denomination, we may call the *fourth root*, that whose biquadrate is equal to a given number; and the *fifth root*, that whose fifth power is equal to a given number, &c. 190. As the square, or second root, is marked by the sign $\sqrt{}$, and the cubic, or third root, by the sign $\sqrt[3]{}$, so the fourth root is represented by the sign $\sqrt[4]{}$; the fifth root by the sign $\sqrt[5]{}$; and so on. It is evident that, according to this method of expression, the sign of the square root ought to be $\sqrt[2]{}$: but as of all roots this occurs most frequently, it has been agreed, for the sake of brevity, to omit the number $2$ as the sign of this root. So that when the radical sign has no num-