Elements of Algebra

CHAP. XI. OF ALGEBRA. 87 The square of $\left\{ \begin{array}{c} \frac{1}{2} \\ \frac{1}{3} \\ \frac{2}{3} \\ \frac{3}{4} \\ \ldots \end{array} \right\}$ is $\left\{ \begin{array}{c} \frac{1}{4} \\ \frac{1}{9} \\ \frac{4}{9} \\ \frac{9}{16} \\ \ldots \end{array} \right\}$ We have only therefore to divide the square of the numerator by the square of the denominator, and the fraction which expresses that division will be the square of the given fraction; thus, $\frac{25}{64}$ is the square of $\frac{5}{8}$; and reciprocally, $\frac{5}{8}$ is the root of $\frac{25}{64}$. 119. When the square of a mixed number, or a number composed of an integer and a fraction, is required, we have only to reduce it to a single fraction, and then take the square of that fraction. Let it be required, for example, to find the square of $2\frac{1}{2}$; we first express this mixed number by $\frac{5}{2}$, and taking the square of that fraction, we have $\frac{25}{4}$, or $6\frac{1}{4}$, for the value of the square of $2\frac{1}{2}$. Also to obtain the square of $3\frac{1}{4}$, we say $3\frac{1}{4}$ is equal to $\frac{13}{4}$; therefore its square is equal to $\frac{169}{16}$, or to $10\frac{9}{16}$. The squares of the numbers between $3$ and $4$, supposing them to increase by one fourth, are as follow: | Numbers. | $3$ | $3\frac{1}{4}$ | $3\frac{1}{2}$ | $3\frac{3}{4}$ | $4$ | |---|---|---|---|---|---| | Squares. | $9$ | $10\frac{9}{16}$ | $12\frac{1}{4}$ | $14\frac{1}{16}$ | $16$ | From this small Table we may infer, that if a root contain a fraction, its square also contains one. Let the root, for example, be $1\frac{1}{2}$; its square is $\frac{289}{144}$, or $2\frac{1}{144}$; that is to say, a little greater than the integer $2$. 120. Let us now proceed to general expressions. First, when the root is $a$, the square must be $aa$; if the root be $2a$, the square is $4aa$; which shews that by doubling the root, the square becomes $4$ times greater; also, if the root be $3a$, the square is $9aa$; and if the root be $4a$, the square is $16aa$. Farther, if the root be $ab$, the square is $aabb$; and if the root be $abc$, the square is $aabbcc$; or $a^2b^2c^2$. 121. Thus, when the root is composed of two, or more factors, we multiply their squares together; and, reciprocally, if a square be composed of two, or more factors, of which each is a square, we have only to multiply together the roots of those squares, to obtain the complete root of the square proposed. Thus, $2304$ is equal to $4 \times 16 \times 36$, the square root of which is $2 \times 4 \times 6$, or $48$; and $48$ is found to be the true square root of $2304$, because $48 \times 48$ gives $2304$. 122. Let us now consider what must be observed on this subject with regard to the signs $+$ and $-$. First, it is 459399