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# Simplify

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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

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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