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# Rationalizing Denominators

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Simplify the following expression to a simplified fraction:

$$\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt5}}.$$

Feb 21, 2021

#1
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First, simplify the terms in the numerator of the fraction.

$$\frac{5}{\sqrt{80}} = \frac{5\sqrt{80}}{80} = \frac{5\cdot4\sqrt{5}}{80}=\frac{\sqrt{5}}{4}$$

$$\frac{\sqrt{845}}{9} = \frac{13\sqrt{5}}{9}$$

$$\sqrt{45} = 3\sqrt{5}$$

The numerator then becomes

$$\frac{\sqrt{5}}{4} + \frac{13\sqrt{5}}{9} + 3\sqrt{5}$$

Since the denominator contains $$\sqrt{5}$$, we can just divide that from the numerator:

$$\frac{1}{4} + \frac{13}{9} + 3 = \frac{169}{36}$$

Now, just take the square root of that to get the final answer:

$$\sqrt{\frac{169}{36}} = \frac{\sqrt{169}}{\sqrt{36}} = \boxed{\frac{13}{6}}$$

Feb 21, 2021
#2
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Hi Textot,

This girl is using a few different identities and she is getting you and others here to do all her homework for her.

Please do not give her full answers like this.   Part answers and good hints are always better.

You can encourage her to interact with you.

You can make up a similar question if you like and show her how to do that one.

There are many options, just please do not give her answers that she can just copy.

Feb 22, 2021