Elements of Algebra

46 ELEMENTS SECT. 1. is $\frac{27}{8}$, or $3\frac{3}{8}$; also the cube of $1\frac{1}{4}$, or of the single fraction $\frac{5}{4}$, is $\frac{125}{64}$, or $1\frac{61}{64}$; and the cube of $3\frac{1}{4}$, or of $\frac{13}{4}$, is $\frac{2197}{64}$, or $34\frac{21}{64}$. 156. Since $aaa$ is the cube of $a$, that of $ab$ will be $aaabbb$; whence we see, that if a number has two or more factors, we may find its cube by multiplying together the cubes of those factors. For example, as $12$ is equal to $3 \times 4$, we multiply the cube of $3$, which is $27$, by the cube of $4$, which is $64$, and we obtain $1728$, the cube of $12$; and farther, the cube of $2a$ is $8aaa$, and consequently $8$ times greater than the cube of $a$: likewise, the cube of $3a$ is $27aaa$; that is to say, $27$ times greater than the cube of $a$. 157. Let us attend here also to the signs $+$ and $-$. It is evident that the cube of a positive number $+a$ must also be positive, that is $+aaa$; but if it be required to cube a negative number $-a$, it is found by first taking the square, which is $+aa$, and then multiplying, according to the rule, this square by $-a$, which gives for the cube required $-aaa$. In this respect, therefore, it is not the same with cubic numbers as with squares, since the latter are always positive: whereas the cube of $-1$ is $-1$, that of $-2$ is $-8$, that of $-3$ is $-27$, and so on. CHAP. XV. Of Cube Roots, and of Irrational Numbers resulting from them. 158. As we can, in the manner already explained, find the cube of a given number, so, when a number is proposed, we may also reciprocally find a number, which, multiplied twice by itself, will produce that number. The number here sought is called, with relation to the other, the cube root; so that the cube root of a given number is the number whose cube is equal to that given number. 159. It is easy therefore to determine the cube root, when the number proposed is a real cube; such as in the examples in the last chapter; for we easily perceive that the cube root of $1$ is $1$; that of $8$ is $2$; that of $27$ is $3$; that of $64$ is $4$, and so on. And, in the same manner, the cube root of $-27$ is $-3$; and that of $-125$ is $-5$. Farther, if the proposed number be a fraction, as $\frac{8}{27}$, the