Elements of Algebra

CHAP. XII. OF ALGEBRA. 41 usual way; thus, $\sqrt{4}$ is the same as $2$; $\sqrt{9}$ is the same as $3$; $\sqrt{36}$ the same as $6$; and $\sqrt{12\frac{1}{4}}$, the same as $\frac{7}{2}$, or $3\frac{1}{2}$. In these instances, the irrationality is only apparent, and vanishes of course. 135. It is easy also to multiply irrational numbers by ordinary numbers; thus, for example, $2$ multiplied by $\sqrt{5}$ makes $2\sqrt{5}$; and $3$ times $\sqrt{2}$ makes $3\sqrt{2}$. In the second example, however, as $3$ is equal to $\sqrt{9}$, we may also express $3$ times $\sqrt{2}$ by $\sqrt{9}$ multiplied by $\sqrt{2}$, or by $\sqrt{18}$; also $2\sqrt{a}$ is the same as $\sqrt{4a}$, and $3\sqrt[3]{a}$ the same as $\sqrt[3]{9a}$; and, in general, $b\sqrt{a}$ has the same value as the square root of $bba$, or $\sqrt{bba}$: whence we infer reciprocally, that when the number which is preceded by the radical sign contains a square, we may take the root of that square, and put it before the sign, as we should do in writing $b\sqrt{a}$ instead of $\sqrt{bba}$. After this, the following reductions will be easily understood: $\left. \begin{array}{l} \sqrt{8}, \text{ or } \sqrt{(2.4)}* \\ \sqrt{12}, \text{ or } \sqrt{(3.4)} \\ \sqrt{18}, \text{ or } \sqrt{(2.9)} \\ \sqrt{24}, \text{ or } \sqrt{(6.4)} \\ \sqrt{32}, \text{ or } \sqrt{(2.16)} \\ \sqrt{75}, \text{ or } \sqrt{(3.25)} \end{array} \right\}$ is equal to $\left\{ \begin{array}{l} 2\sqrt{2} \\ 2\sqrt{3} \\ 3\sqrt{2} \\ 2\sqrt{6} \\ 4\sqrt{2} \\ 5\sqrt{3} \end{array} \right.$ and so on. 136. Division is founded on the same principles; as $\sqrt{a}$ divided by $\sqrt{b}$ gives $\frac{\sqrt{a}}{\sqrt{b}}$, or $\sqrt{\frac{a}{b}}$. In the same manner, $\left. \begin{array}{l} \frac{\sqrt{8}}{\sqrt{2}} \\ \frac{\sqrt{18}}{\sqrt{2}} \\ \frac{\sqrt{12}}{\sqrt{3}} \end{array} \right\}$ is equal to $\left\{ \begin{array}{l} \sqrt{\frac{8}{2}}, \text{ or } \sqrt{4}, \text{ or } 2 \\ \sqrt{\frac{18}{2}}, \text{ or } \sqrt{9}, \text{ or } 3 \\ \sqrt{\frac{12}{3}}, \text{ or } \sqrt{4}, \text{ or } 2. \end{array} \right.$ Farther, $\left. \begin{array}{l} \frac{2}{\sqrt{2}} \\ \frac{3}{\sqrt{3}} \\ \frac{12}{\sqrt{6}} \end{array} \right\}$ is equal to $\left\{ \begin{array}{l} \frac{\sqrt{4}}{\sqrt{2}}, \text{ or } \sqrt{\frac{4}{2}}, \text{ or } \sqrt{2}, \\ \frac{\sqrt{9}}{\sqrt{3}}, \text{ or } \sqrt{\frac{9}{3}}, \text{ or } \sqrt{3}, \\ \frac{\sqrt{144}}{\sqrt{6}}, \text{ or } \sqrt{\frac{144}{6}}, \text{ or } \sqrt{24}, \end{array} \right.$ or $\sqrt{(6 \times 4)}$, or lastly $2\sqrt{6}$. 137. There is nothing in particular to be observed in ad- * The point between 2.4, 3.4, \ldots indicates multiplication.