Is \sqrt{3-2\sqrt{2}}+1 = \sqrt{2} ? If so, why? How can it be proven?
D dmouthfan2028 New member Joined Feb 18, 2010 Messages 3 Apr 1, 2010 #1 Is 3−22+1=2\displaystyle \sqrt{3-2\sqrt{2}}+1 = \sqrt{2}3−22+1=2 ? If so, why? How can it be proven?
Is 3−22+1=2\displaystyle \sqrt{3-2\sqrt{2}}+1 = \sqrt{2}3−22+1=2 ? If so, why? How can it be proven?
S soroban Elite Member Joined Jan 28, 2005 Messages 5,586 Apr 1, 2010 #2 Hello, dmouthfan2028! Is 3−22+1 = 2 ?\displaystyle \text{Is }\sqrt{3-2\sqrt{2}}+1 \:=\: \sqrt{2}\text{ ?}Is 3−22+1=2 ? If so, why? How can it be proven?\displaystyle \text{If so, why? }\:\text{ How can it be proven?}If so, why? How can it be proven? Click to expand... We have: x = 3−22+1\displaystyle \text{We have: }\:x \;=\;\sqrt{3-2\sqrt{2}} + 1We have: x=3−22+1 Note that: 3−22 = (2−1)2\displaystyle \text{Note that: }3- 2\sqrt{2} \;=\;(\sqrt{2}-1)^2Note that: 3−22=(2−1)2 Hence: 3−22+1 = (2−1)2+1 = (2−1)+1 = 2\displaystyle \text{Hence: }\:\sqrt{3-2\sqrt{2}} + 1 \;=\;\sqrt{(\sqrt{2}-1)^2} + 1 \;=\;(\sqrt{2}-1) + 1 \;=\;\sqrt{2}Hence: 3−22+1=(2−1)2+1=(2−1)+1=2
Hello, dmouthfan2028! Is 3−22+1 = 2 ?\displaystyle \text{Is }\sqrt{3-2\sqrt{2}}+1 \:=\: \sqrt{2}\text{ ?}Is 3−22+1=2 ? If so, why? How can it be proven?\displaystyle \text{If so, why? }\:\text{ How can it be proven?}If so, why? How can it be proven? Click to expand... We have: x = 3−22+1\displaystyle \text{We have: }\:x \;=\;\sqrt{3-2\sqrt{2}} + 1We have: x=3−22+1 Note that: 3−22 = (2−1)2\displaystyle \text{Note that: }3- 2\sqrt{2} \;=\;(\sqrt{2}-1)^2Note that: 3−22=(2−1)2 Hence: 3−22+1 = (2−1)2+1 = (2−1)+1 = 2\displaystyle \text{Hence: }\:\sqrt{3-2\sqrt{2}} + 1 \;=\;\sqrt{(\sqrt{2}-1)^2} + 1 \;=\;(\sqrt{2}-1) + 1 \;=\;\sqrt{2}Hence: 3−22+1=(2−1)2+1=(2−1)+1=2
