Simplify :
2223+1+1723−1\dfrac{22}{2\sqrt{3} + 1} + \dfrac{17}{2\sqrt{3} - 1}23+122+23−117
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(i) Solving,
⇒2223+1+1723−1⇒22(23−1)+17(23+1)(23+1)(23−1)⇒443−22+343+17(23)2−12⇒783−512−1⇒783−511.\Rightarrow \dfrac{22}{2\sqrt{3} + 1} + \dfrac{17}{2\sqrt{3} - 1} \\[1em] \Rightarrow \dfrac{22(2\sqrt{3} - 1) + 17(2\sqrt{3} + 1)}{(2\sqrt{3} + 1)(2\sqrt{3} - 1)} \\[1em] \Rightarrow \dfrac{44\sqrt{3} - 22 + 34\sqrt{3} + 17}{(2\sqrt{3})^2 - 1^2} \\[1em] \Rightarrow \dfrac{78\sqrt{3} - 5}{12 - 1} \\[1em] \Rightarrow \dfrac{78\sqrt{3} - 5}{11}.⇒23+122+23−117⇒(23+1)(23−1)22(23−1)+17(23+1)⇒(23)2−12443−22+343+17⇒12−1783−5⇒11783−5.
Hence, solution = 783−511.\dfrac{78\sqrt{3} - 5}{11}.11783−5.
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If x = 5−25+2 and y=5+25−2\dfrac{\sqrt{5} - 2}{\sqrt{5} + 2} \text{ and } y = \dfrac{\sqrt{5} + 2}{\sqrt{5} - 2}5+25−2 and y=5−25+2; find :
(i) x2
(ii) y2
(iii) xy
(iv) x2 + y2 + xy
