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Simplify the following expression to a simplified fraction:
$$\sqrt{\dfrac{\dfrac{5}{\sqrt{40}}+\dfrac{\sqrt{845}}{18}+\sqrt{45}}{\sqrt10}}.$$

 Jun 7, 2022
 #1
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The first step is to simplify the numerator of the fraction. 

 

\(\large{{5 \over \sqrt {40}} = {5 \over {2 \sqrt {10}}} = {10 \sqrt{10} \over 40} = {\sqrt {10} \over 4}}\)

\(\large{{ \sqrt {845} \over 18} = {13 \sqrt 5 \over 18}}\)

\(\sqrt {45} = \sqrt 9 \times \sqrt 5 = 3\sqrt 5\)

 

Now, we have to add these up. Focusing on the final 2 fractions, we can simplify the third fraction to \({54 \sqrt 5 \over 18}\). Adding the 2 gives us: \({67 \sqrt 5 \over 18}\)

 

To add the first fraction, we simplify each so it has a common denominator, then add to get: \(\large{{9 \sqrt {10} \over 36} + {134\sqrt{5} \over 36} = { 134 \sqrt 5 + 9 \sqrt {10} \over 36}}\)

 

Now, we need to rationalize the denominator of: \(\large{{{134 \sqrt 5 + 9 \sqrt{10} \over 36} \over \sqrt{10}}}\)

 

Multiplying by \(\sqrt {10} \over \sqrt {10}\), we get: \({{134 \sqrt {50} + 90 \over 36} \over 10} = {134 \sqrt {50} + 90 \over 360} = {67 \sqrt {50} + 45 \over 180}\)

 

This means our final answer is: \(\color{brown}\boxed{67 \sqrt {50} + 45 \over 180}\)

 Jun 8, 2022

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