# How do you find the square root of 38.94?

Aug 5, 2016

$= 6.24$

#### Explanation:

$\sqrt{38.94}$
$= 6.24$

Aug 5, 2016

You find it numerically. The actual square root is $\approx 6.2402$.

The pattern with squares to get ${\left(n + 1\right)}^{2}$ is to add the sum of $n$ and $n + 1$ onto ${n}^{2}$:

$\textcolor{b l u e}{{\left(n + 1\right)}^{2}} = {n}^{2} + 2 n + 1$

$= \textcolor{b l u e}{{n}^{2} + \left[\left(n\right) + \left(n + 1\right)\right]}$

1. Since ${60}^{2} = 3600$, ${61}^{2} = {60}^{2} + 60 + 61 = 3721$, and ${62}^{2} = {61}^{2} + 61 + 62 = 3844$.

2. Since $3894 - 3844 < n$, $3894$ is not a perfect square. Furthermore, since $\frac{3894}{100} = 38.94$, and $100 = {10}^{2}$, if $3894$ has no perfect square, neither does $38.94$. Therefore, it has an irregular decimal answer.

3. From the above pattern, ${63}^{2} = {62}^{2} + 62 + 63 = 3969$, and $3894$ is 40% of the way from $3844$ to $3969$.

4. Since ${x}^{2}$ is approximately linear at high $x$, we can estimate the square root of $38.94$ to be about $\left(62 + 0.4\right) \times \frac{1}{10} = \textcolor{b l u e}{6.24}$.

The actual square root is $\approx 6.2402$.