Simplify the following expression to a simplified fraction:
$$\sqrt{\dfrac{\dfrac{5}{\sqrt{40}}+\dfrac{\sqrt{845}}{18}+\sqrt{45}}{\sqrt10}}.$$
+2666 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: \(\color{brown}\boxed{67 \sqrt {50} + 45 \over 180}\)
