# How do you calculate 4sqrt(5) ?

Oct 7, 2017

See below.

#### Explanation:

$4 \sqrt{5}$ is also expressed as $4$ times $\sqrt{5}$, or $4 \cdot \sqrt{5}$.

If you were to type $\sqrt{5}$ in your calculator, you would get a value of $2.2360679775$. Then, take this and multiply it by $4$ to get the final answer, $8.94427191$.

I hope that helps!

Oct 7, 2017

$8.9442$

#### Explanation:

$4 \sqrt{5} = 4 \cdot 2.2361 = 8.9442$

Oct 7, 2017

$4 \sqrt{5} \approx \frac{51841}{5796} \approx 8.94427191$

#### Explanation:

Note that ${\left(4 \sqrt{5}\right)}^{2} = {4}^{2} \cdot 5 = 16 \cdot 5 = 80 < 81 = {9}^{2}$

So $4 \sqrt{5}$ is an irrational number a little less than $9$.

Note that in general:

$\sqrt{{a}^{2} + b} = a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \ldots}}}}$

So we can put $a = 9$ and $b = - 1$ to find:

$4 \sqrt{5} = \sqrt{80} = \sqrt{{9}^{2} - 1} = 9 - \frac{1}{18 - \frac{1}{18 - \frac{1}{18 - \frac{1}{18 - \ldots}}}}$

We can terminate this generalised continued fraction early to get rational approximations to $4 \sqrt{5}$.

For example:

$4 \sqrt{5} \approx 9 - \frac{1}{18} = 8.9 \overline{4}$

$4 \sqrt{5} \approx 9 - \frac{1}{18 - \frac{1}{18}} = 9 - \frac{18}{323} = \frac{2889}{323} \approx 8.944272$

$4 \sqrt{5} \approx 9 - \frac{1}{18 - \frac{1}{18 - \frac{1}{18}}} = 9 - \frac{1}{18 - \frac{18}{323}} = 9 - \frac{323}{5796} = \frac{51841}{5796} \approx 8.94427191$