# How do you find the square root of 50?

Jun 7, 2016

$\sqrt{50} = 5 \sqrt{2}$

#### Explanation:

$\sqrt{50}$

= $\sqrt{2 \times \underline{5 \times 5}}$

= $5 \sqrt{2}$

Jun 8, 2016

$\sqrt{50}$ can be simplified as $5 \sqrt{2}$

We can also find rational approximations to it.

For example:

$\sqrt{50} \approx 7 \frac{14}{197} \approx 7.071066$

#### Explanation:

The square root of $50$ is not a whole number, or even a rational number. It is an irrational number, but you can simplify it or find rational approximations for it.

First note that

$50 = 2 \times 5 \times 5$

contains a square factor ${5}^{2}$. We can use this to simplify the square root:

$\sqrt{50} = \sqrt{{5}^{2} \cdot 2} = \sqrt{{5}^{2}} \cdot \sqrt{2} = 5 \sqrt{2}$

Apart from simplifying it algebraically, what is its numerical value?

Note that ${7}^{2} = 49$, so $\sqrt{49} = 7$ and $\sqrt{50}$ will be slightly larger than $7$.

In fact, since $50 = {7}^{2} + 1$, the square root of $50$ is expressible as a very regular continued fraction:

$\sqrt{50} = 7 + \frac{1}{14 + \frac{1}{14 + \frac{1}{14 + \frac{1}{14 + \frac{1}{14 + \frac{1}{14 + \ldots}}}}}}$

This can be written as sqrt(50) = [7;bar(14)] where the bar over the $14$ indicates the repeating part of the continued fraction.

We can terminate the continued fraction early to give us rational approximations for $\sqrt{50}$.

For example:

sqrt(50) ~~ [7;14] = 7+1/14 = 7.0bar(714285)

sqrt(50) ~~ [7;14,14] = 7+1/(14+1/14) = 7+14/197 ~~ 7.071066

In fact:

$\sqrt{50} \approx 7.071067811865475244$