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Other answers:

I have to draw it lol
you're fine, no worries
that's definitely the best option instead of describing it in words
It isn't working. I have no idea what to do.
ok one sec
OK my drawing sucks but this is it.
thanks, and your drawing is perfect, no worries
you need to rationalize the denominator, so you need to multiply top and bottom by \[\Large 10\sqrt{2} + \sqrt{10}\]
Doing so will give you \[\Large \frac{5\sqrt{2}+\sqrt{10}}{10\sqrt{2} - \sqrt{10}}\] \[\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{(10\sqrt{2} - \sqrt{10})(10\sqrt{2} + \sqrt{10})}\] Do you know what to do from here?
No this is where I got lost.
ok you would use the difference of squares rule to expand out the denominator \[\Large (10\sqrt{2} - \sqrt{10})(10\sqrt{2} + \sqrt{10})\] \[\Large (10\sqrt{2})^2 - (\sqrt{10})^2\] \[\Large 10^2*(\sqrt{2})^2 - (\sqrt{10})^2\] \[\Large 100*2 - 10\] \[\Large 200 - 10\] \[\Large 190\]
So \[\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{(10\sqrt{2} - \sqrt{10})(10\sqrt{2} + \sqrt{10})}\] turns into \[\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{190}\]
You just need to expand out the numerator, then you're done
Doing that will give you \[\Large (5\sqrt{2}+\sqrt{10})(10\sqrt{2} + \sqrt{10})\] \[\Large 5\sqrt{2}(10\sqrt{2} + \sqrt{10})+\sqrt{10}(10\sqrt{2} + \sqrt{10})\] \[\Large 5\sqrt{2}*10\sqrt{2} + 5\sqrt{2}*\sqrt{10}+\sqrt{10}*10\sqrt{2} + \sqrt{10}*\sqrt{10}\] \[\Large 100 + 5\sqrt{20}+10\sqrt{20} + 10\] \[\Large 110 + 15\sqrt{20}\] \[\Large 110 + 15\sqrt{4*5}\] \[\Large 110 + 15\sqrt{4}*\sqrt{5}\] \[\Large 110 + 15*2*\sqrt{5}\] \[\Large 110 + 30*\sqrt{5}\]
So \[\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{190}\] turns into \[\Large \frac{110 + 30*\sqrt{5}}{190}\] I guess from here you can divide each term by 10 to get \[\Large \frac{11 + 3\sqrt{5}}{19}\] and you're done
So this shows us that \[\Large \frac{5\sqrt{2}+\sqrt{10}}{10\sqrt{2} - \sqrt{10}} =\frac{11 + 3\sqrt{5}}{19}\]
Neat. You explained it so well! I wouldn't have been able to foil that like you did. Thanks!
you're welcome

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