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Simplify the following expression to a simplified fraction: \(\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt5}}\)

 Jun 12, 2021
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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!

 Jun 12, 2021
edited by xCorrosive  Jun 12, 2021

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