+100 Simplify the following expression to a simplified fraction: \(\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt5}}\)
+179 Let's take the inside first and rationalize the denominator. (note that the outer square root still exists, it's just more convenient this way).
$$\frac{ \sqrt{5} (\frac{5}{\sqrt{80}} + \frac{ \sqrt{845}}{9} + \sqrt{45})}{\sqrt{5} \cdot \sqrt{5}}$$.
Simplifying the fraction gives
$$\frac{\frac{5 \sqrt{5}}{\sqrt{80}} + \frac{65}{9} + 15}{5}$$
Simplifying the fraction further gives
$$\frac{\frac{5}{4} + \frac{65}{9} + 15}{5}$$
Doing some basic operations (adding and dividing) gives us
$$ \frac{169}{36} $$
Finally, we finish by applying the outer square root. This gives us
$$ \sqrt{\frac{169}{36}} = \boxed{\frac{13}{6}} $$,
which is our final answer. The $\LaTeX$ on that one was tough!
