Square Roots of Compound Expressions.

Therefore, the number of decimal places in every square decimal will be even, and the number of decimal places in the root will be half as many as in the given number itself.

Hence, if a given number contain a decimal, we divide it into groups of two figures each, by beginning at the decimal point and marking toward the left for the integral number, and toward the right for the decimal. We must have the last group on the right of the decimal point contain two figures, annexing a cipher when necessary.

Ex. Find the square roots of $41.2164$ and $965.9664$.

$$\begin{gathered} r@r@l@ll\&41\&\mathrm{.21}\&64\&(6.42 \\ \&36 \\ \mathrm{\backslash cline}2-3124\text{)}\&5\&21 \\ \&4\&96 \\ \mathrm{\backslash cline}3-4\mathrm{\backslash multicolumn}2r1282\text{)}\&\mathrm{\backslash multicolumn}2l2564 \\ \&\&\mathrm{\backslash multicolumn}2l2564 \\ \mathrm{\backslash cline}3-4rl@l@ll9\&65\&\mathrm{.96}\&64\&(31.08 \\ 9 \\ \mathrm{\backslash cline}2-261\text{)}\&65 \\ \&61 \\ \mathrm{\backslash cline}2-36208\text{)}\&49\&664 \\ \&49\&664 \\ \mathrm{\backslash cline}2-3 \end{gathered}$$

If a number contain an odd number of decimal places, or if any number give a remainder when as many figures in the root have been obtained as the given number has groups, then its exact square root cannot be found. We may, however, approximate to its exact root as near as we please by annexing ciphers and continuing the operation.

The square root of a common fraction whose denominator is not a perfect square can be found approximately by reducing the fraction to a decimal and then extracting the root; or by reducing the fraction to an equivalent fraction whose denominator is a perfect square, and extracting the square root of both terms of the fraction.

1. Find the square roots of $3$ and $357.357$.

$rcccl\&3.\&\mathrm{\backslash multicolumn}3l(\text{{}$1.732\dots$$\begin{gathered} \}\} \\ \&1 \\ \mathrm{\backslash cline}2-327\text{)}\&2\&00 \\ \&1\&89 \\ \mathrm{\backslash cline}3-4\mathrm{\backslash multicolumn}2r343\text{)}\&11\&00 \\ \&\&10\&29 \\ \mathrm{\backslash cline}4-5\mathrm{\backslash multicolumn}3r3462\text{)}\&71\&00 \\ \&\&\&69\&24 \\ \mathrm{\backslash cline}4-5rcclll\&3\&57.\&35\&70\&\text{{} \end{gathered}$$(18.903\dots$$\begin{gathered} \} \\ \&1 \\ \mathrm{\backslash cline}2-328\text{)}\&2\&57 \\ \&2\&24 \\ \mathrm{\backslash cline}3-4\mathrm{\backslash multicolumn}2r369\text{)}\&33\&35 \\ \&\&33\&21 \\ \mathrm{\backslash cline}4-6\mathrm{\backslash multicolumn}3r37803\text{)}\&14\&70\&00 \\ \&\&\&11\&34\&09 \\ \mathrm{\backslash cline}4-6 \end{gathered}$$

2. Find the square root of $\frac{5}{8}$.

lint Since 58 = 0.625, lintthe square root of 58 = 0.625 = 0.79057. [1] lint Or, 58 = 1016, lintand the square root of 58 = 1016 = 1410 = 14(3.16227) = 0.79057.

Exercise 79.

Find the square root of:

1. $324$.

2. $441$.

3. $529$.

4. $961$.

5. $10.24$.

6. $53.29$.