# How do you find the square root of 6724 ?

May 3, 2017

$\sqrt{6724} = 82$

#### Explanation:

We can try factoring $6724$ and identifying square factors:

$\textcolor{w h i t e}{000000} 6724$
$\textcolor{w h i t e}{000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{00000} 2 \textcolor{w h i t e}{000} 3362$
$\textcolor{w h i t e}{000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{00000000} 2 \textcolor{w h i t e}{000} 1681$
$\textcolor{w h i t e}{000000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{00000000000} 41 \textcolor{w h i t e}{00} 41$

So:

$\sqrt{6724} = \sqrt{{2}^{2} \cdot {41}^{2}} = 2 \cdot 41 = 82$

That "worked", but I cheated slightly in knowing that $1681 = {41}^{2}$

Let's try another approach...

Given $6724$, split off pairs of digits starting from the right to get:

$67 | 24$

Looking at the most significant pair of digits, note that:

$67 > 64 = {8}^{2}$

Hence the square root of $6724$ is a little larger than $80$

We can use the Babylonian method to find a better approximation:

Given a number $n$ of which you want the square root and an approximation ${a}_{i}$ to that root, a better approximation ${a}_{i + 1}$ is given by the formula:

${a}_{i + 1} = \frac{{a}_{i}^{2} + n}{2 {a}_{i}}$

So in our case, with $n = 6724$ and ${a}_{0} = 80$ we find:

${a}_{1} = \frac{{80}^{2} + 6724}{2 \cdot 80} = \frac{6400 + 6724}{160} = 82.025$

Hmmm - that's suspiciously close to $82$. Does $82$ work?

${82}^{2} = {\left(80 + 2\right)}^{2} = {80}^{2} + 2 \cdot 80 \cdot 2 + {2}^{2} = 6400 + 320 + 4 = 6724$