# Simplifying Order of Operations [closed]

I am familiar with the PEMDAS rule, but how would we simplify something as complex as this:

$$\frac{\frac{3}{4}+2\frac{\sqrt{6}}{5}}{1-\frac{3}{4}(2\frac{\sqrt{6}}{5})}$$

Can we work on the top side first and then move to the bottom? I tried doing that but my result did not match the correct answer that was provided by my book, so either I made a mistake or what I did was outright wrong.

## closed as unclear what you're asking by Rory Daulton, gebruiker, kingW3, Arnaldo, C. FalconMay 21 '17 at 0:00

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• Does your question have anything to do with the partial solution shown at your link, or are you using only the question itself? And what exactly did you try--your explanation is not clear. – Rory Daulton May 20 '17 at 10:42

Yes, you can work the top and then the bottom. But it's probably less work, in these "four-story fraction" problems, to clear the inside denominators. In this case, $20$ is a common denominator of all the little fractions, so multiply top and bottom by $20$ to get

$$\frac{15+8\sqrt{6}}{20 - 6\sqrt{6}}.$$

Often "simplify" includes rationalizing the denominator. If you multiply top and bottom by $20 + 6\sqrt{6}$ you get

$$\frac{15+8\sqrt{6}}{20 - 6\sqrt{6}}\frac{20+6\sqrt{6}}{20+6\sqrt{6}} = \frac{ 300 +160\sqrt{6}+90\sqrt{6} +48\cdot 6}{400-36\cdot 6}$$

$$= \frac{588+250\sqrt{6}}{184} = \frac{294+125\sqrt{6}}{92}.$$

Where, at last step, we divided top and bottom by $2$. That's about as simple as this one gets.

• You wrote $150\sqrt{6}$ when I think you meant $160\sqrt{6}$. – John Wayland Bales May 20 '17 at 18:17
• So the correct answer is $\dfrac{294+125\sqrt{6}}{92}$. – John Wayland Bales May 22 '17 at 6:01

we have $$1-\frac{3}{4}\cdot 2\frac{\sqrt{6}}{5}=1-{3}\frac{\sqrt{6}}{10}=\frac{10-3\sqrt{6}}{10}$$ and $$\frac{3}{4}+\frac{2}{5}\sqrt{6}=\frac{15+8\sqrt{6}}{20}$$