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 April 21st, 2011, 10:56 PM #1 Member   Joined: Feb 2009 Posts: 54 Thanks: 0 Can't simplify When I type $\sqrt{7-2\cdot\sqrt{10}} + \sqrt2$ I get $\sqrt5$ as an answer. Yet I can't simplify it on paper. Either I get many pages of transformations and come back to the start or I go far away into the wrong weird big answers. I tried to multiply $\frac{\sqrt{7-2\cdot\sqrt{10}} + \sqrt2}{1}\cdot{\sqrt{7-2\cdot\sqrt{10}} - \sqrt2}$ and then multiply denominator $a+b$ by $a-b$ until roots in it disappear but I come back to the start like I said above. Looks like I'm walking in circles. This is a part of a bigger polynomial which my calculator can solve but I can't. There must be some trick my calculator knows but I don't.
 April 21st, 2011, 11:04 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Can't simplify We have: $\sqrt{7-2\sqrt{10}}+\sqrt{2}$ We need to rewrite the first radicand: $7-2\sqrt{10}=5-2\sqrt{10}+2=$$\sqrt{5}$$^2-2\sqrt{5}\sqrt{2}+$$\sqrt{2}$$^2=$$\sqrt{5}-\sqrt{2}$$^2$ This allows us to write: $\sqrt{7-2\sqrt{10}}+\sqrt{2}=\sqrt{$$\sqrt{5}-\sqrt{2}$$^2}+\sqrt{2}=\sqrt{5}-\sqrt{2}+\sqrt{2}=\sqrt{5}$
 April 21st, 2011, 11:18 PM #3 Member   Joined: Feb 2009 Posts: 54 Thanks: 0 Re: Can't simplify Okay that was surprising. And that calculator can do that
 April 21st, 2011, 11:27 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Can't simplify What was essential was the recognition that 5 + 2 = 7 and 5·2 = 10. If you have: $a\pm2\sqrt{c}$ and d + e = a and d·e = c then $a\pm2\sqrt{c}=$$\sqrt{d}\pm\sqrt{e}$$^2$

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