# What is the square root of 256.25 ?

Dec 17, 2016

$\sqrt{256 \cdot 25} = 80$

$\sqrt{256.25} \approx 16.00781$

#### Explanation:

Let's look at both possibilities:

$\textcolor{w h i t e}{}$
Square root of $256 \cdot 25$

If there is a typo in the question and the decimal point was meant to signify multiplication then we find:

$\sqrt{256 \cdot 25} = \sqrt{{16}^{2} \cdot {5}^{2}} = \sqrt{{16}^{2}} \cdot \sqrt{{5}^{2}} = 16 \cdot 5 = 80$

$\textcolor{w h i t e}{}$
Square root of $256.25$

If the question is correct as stands, then we cannot simplify the square root, but we can approximate it.

Note that $4 \cdot 256.25 = 1025 = {32}^{2} + 1$

Since this is of the form ${n}^{2} + 1$ its square root is expressible as very regular continued fraction:

sqrt(1025) = [32;bar(64)] = 32+1/(64+1/(64+1/(64+1/(64+...))))

Hence:

sqrt(256.25) = 1/2 sqrt(1025) = 1/2 [32;bar(64)]

We can truncate the continued fraction to get approximations.

For example:

$\sqrt{256.25} \approx \frac{1}{2} \left[32 , 64\right] = \frac{1}{2} \left(32 + \frac{1}{64}\right) = 16 + \frac{1}{128} = 16.0078125 \approx 16.00781$