Square Roots of Compound Expressions.

of groups of figures. The last group to the left may have only one figure.

Ex. Find the square root of $3249$.

$$\begin{gathered} rc@ll\&3\&249\&(57 \\ \&2\&5 \\ \mathrm{\backslash cline}2-3107\text{)}\&\&749 \\ \&\&749 \\ \mathrm{\backslash cline}3-3 \end{gathered}$$

In this case, $a$ in the typical form $a^2+2ab+b^2$ represents $5$ tens, that is, $50$, and $b$ represents $7$. The $25$ subtracted is really $2500$, that is, $a^2$, and the complete divisor $2a+b$ is $2\times 50+7=107$.

The same method will apply to numbers of more than two groups of figures by considering $a$ in the typical form to represent at each step the part of the root already found.

It must be observed that $a$ represents so many tens with respect to the next figure of the root.

Ex. Find the square root of $94,249$.

$$\begin{gathered} r*3c@l\&9\&42\&49\&(307 \\ \&9 \\ \mathrm{\backslash cline}2-4607\text{)}\&\&\mathrm{\backslash multicolumn}2l4249 \\ \&\&\mathrm{\backslash multicolumn}2l4249 \\ \mathrm{\backslash cline}3-4 \end{gathered}$$

Since the first trial divisor, $60$, is not contained in $42$, we put a zero in the root, and bring down the next group, $49$.

If the square root of a number has decimal places, the number itself will have twice as many. Thus, if $0.21$ is the square root of some number, this number will be $(0.21{)}^2=0.21\times 0.21=0.0441$, and if $0.111$ be the root, the number will be $(0.111{)}^2=0.111x0.111=0.012321$.