## julia_copen 2 years ago Help with this last radical?

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1. jim_thompson5910

2. julia_copen

Hang on a second

3. jim_thompson5910

ok

4. julia_copen

I have to draw it lol

5. jim_thompson5910

you're fine, no worries

6. jim_thompson5910

that's definitely the best option instead of describing it in words

7. julia_copen

It isn't working. I have no idea what to do.

8. julia_copen

|dw:1364269301720:dw|

9. jim_thompson5910

ok one sec

10. julia_copen

OK my drawing sucks but this is it.

11. jim_thompson5910

thanks, and your drawing is perfect, no worries

12. jim_thompson5910

you need to rationalize the denominator, so you need to multiply top and bottom by $\Large 10\sqrt{2} + \sqrt{10}$

13. jim_thompson5910

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?

14. julia_copen

No this is where I got lost.

15. jim_thompson5910

ok you would use the difference of squares rule to expand out the denominator $\Large (10\sqrt{2} - \sqrt{10})(10\sqrt{2} + \sqrt{10})$ $\Large (10\sqrt{2})^2 - (\sqrt{10})^2$ $\Large 10^2*(\sqrt{2})^2 - (\sqrt{10})^2$ $\Large 100*2 - 10$ $\Large 200 - 10$ $\Large 190$

16. jim_thompson5910

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 $\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{190}$

17. jim_thompson5910

You just need to expand out the numerator, then you're done

18. jim_thompson5910

Doing that will give you $\Large (5\sqrt{2}+\sqrt{10})(10\sqrt{2} + \sqrt{10})$ $\Large 5\sqrt{2}(10\sqrt{2} + \sqrt{10})+\sqrt{10}(10\sqrt{2} + \sqrt{10})$ $\Large 5\sqrt{2}*10\sqrt{2} + 5\sqrt{2}*\sqrt{10}+\sqrt{10}*10\sqrt{2} + \sqrt{10}*\sqrt{10}$ $\Large 100 + 5\sqrt{20}+10\sqrt{20} + 10$ $\Large 110 + 15\sqrt{20}$ $\Large 110 + 15\sqrt{4*5}$ $\Large 110 + 15\sqrt{4}*\sqrt{5}$ $\Large 110 + 15*2*\sqrt{5}$ $\Large 110 + 30*\sqrt{5}$

19. jim_thompson5910

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

20. jim_thompson5910

So this shows us that $\Large \frac{5\sqrt{2}+\sqrt{10}}{10\sqrt{2} - \sqrt{10}} =\frac{11 + 3\sqrt{5}}{19}$

21. julia_copen

Neat. You explained it so well! I wouldn't have been able to foil that like you did. Thanks!

22. jim_thompson5910

you're welcome