Evaluate :
4−54+5+4+54−5\dfrac{4 - \sqrt{5}}{4 + \sqrt{5}} + \dfrac{4 + \sqrt{5}}{4 - \sqrt{5}}4+54−5+4−54+5
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Solving,
⇒4−54+5+4+54−5⇒(4−5)2+(4+5)2(4+5)(4−5)⇒42+(5)2−2×4×5+42+(5)2+2×4×542−(5)2⇒16+5−85+16+5+8516−5⇒4211⇒3911.\Rightarrow \dfrac{4 - \sqrt{5}}{4 + \sqrt{5}} + \dfrac{4 + \sqrt{5}}{4 - \sqrt{5}} \\[1em] \Rightarrow \dfrac{(4 - \sqrt{5})^2 + (4 + \sqrt{5})^2}{(4 + \sqrt{5})(4 - \sqrt{5})} \\[1em] \Rightarrow \dfrac{4^2 + (\sqrt{5})^2 - 2 \times 4 \times \sqrt{5} + 4^2 + (\sqrt{5})^2 + 2 \times 4 \times \sqrt{5}}{4^2 - (\sqrt{5})^2} \\[1em] \Rightarrow \dfrac{16 + 5 - 8\sqrt{5} + 16 + 5 + 8\sqrt{5}}{16 - 5} \\[1em] \Rightarrow \dfrac{42}{11} \\[1em] \Rightarrow 3\dfrac{9}{11}.⇒4+54−5+4−54+5⇒(4+5)(4−5)(4−5)2+(4+5)2⇒42−(5)242+(5)2−2×4×5+42+(5)2+2×4×5⇒16−516+5−85+16+5+85⇒1142⇒3119.
Hence, 4−54+5+4+54−5=3911\dfrac{4 - \sqrt{5}}{4 + \sqrt{5}} + \dfrac{4 + \sqrt{5}}{4 - \sqrt{5}} = 3\dfrac{9}{11}4+54−5+4−54+5=3119.
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(ii) 13+22\dfrac{1}{3 + 2\sqrt{2}}3+221
(iii) 2−33\dfrac{2 - \sqrt{3}}{\sqrt{3}}32−3
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Since 90 = 2 x 3 x 3 x 5, 2390\dfrac{23}{90}9023 is not a terminating decimal;
True
False
none of these
