# How do you find the square root of 1849?

Jul 21, 2018

$\sqrt{1849} = 43$

#### Explanation:

We could first seek to find the prime factorisation of $1849$, but as we shall see it is actually the square of a prime number, so that would be somewhat tedious.

Alternatively, let's split it into pairs of digits from the right to get:

$18 \text{|} 49$

Examining the leading $18$, note that it lies between ${4}^{2}$ and ${5}^{2}$:

${4}^{2} = 16 < 18 < 25 = {5}^{2}$

So:

$4 < \sqrt{18} < 5$

and hence:

$40 < \sqrt{1849} < 50$

To find a suitable correction, we can linearly interpolate between $40$ and $50$ to find:

$\sqrt{1849} \approx 40 + \left(50 - 40\right) \cdot \frac{1849 - {40}^{2}}{{50}^{2} - {40}^{2}}$

$\textcolor{w h i t e}{\sqrt{1849}} \approx 40 + 10 \cdot \frac{1849 - 1600}{2500 - 1600}$

$\textcolor{w h i t e}{\sqrt{1849}} \approx 40 + \frac{2490}{900}$

$\textcolor{w h i t e}{\sqrt{1849}} \approx 40 + 2.49 + 0.249 + 0.0249 + \ldots$

$\textcolor{w h i t e}{\sqrt{1849}} \approx 42.76$

Hmmm... That's close to $43$, let's try ${43}^{2}$...

$43 \cdot 43 = {40}^{2} + 2 \cdot 40 \cdot 3 + {3}^{2} = 1600 + 240 + 9 = 1849$

So:

$\sqrt{1849} = 43$