Ugly expression?

Algebra Level 4

( 5 − 2 5 ) ( 17 − 17 ) ( 5 + 2 5 ) ( 17 + 17 ) + ( 5 − 2 5 ) ( 85 − 2 85 ) ( 5 + 2 5 ) ( 85 + 2 85 ) + ( 17 − 17 ) ( 85 − 2 85 ) ( 17 + 17 ) ( 85 + 2 85 ) \sqrt{\frac{\left(5-2 \sqrt{5}\right) \left(17-\sqrt{17}\right)}{\left(5+2 \sqrt{5}\right) \left(17+\sqrt{17}\right)}}+\sqrt{\frac{\left(5-2 \sqrt{5}\right) \left(85-2 \sqrt{85}\right)}{\left(5+2 \sqrt{5}\right) \left(85+2 \sqrt{85}\right)}}+\sqrt{\frac{\left(17-\sqrt{17}\right) \left(85-2 \sqrt{85}\right)}{\left(17+\sqrt{17}\right) \left(85+2 \sqrt{85}\right)}}

Evaluate the expression above, give your answer to 3 decimal places.


The answer is 1.

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Alan Enrique Ontiveros Salazar by Alan Enrique Ontiveros S… , 22

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1 solution

Chew-Seong Cheong
Sep 26, 2015

( 5 − 2 5 ) ( 17 − 17 ) ( 5 + 2 5 ) ( 17 + 17 ) + ( 5 − 2 5 ) ( 85 − 2 85 ) ( 5 + 2 5 ) ( 85 + 2 85 ) + ( 17 − 17 ) ( 85 − 2 85 ) ( 17 + 17 ) ( 85 + 2 85 ) = 5 ( 5 − 2 ) 17 ( 17 − 1 ) 5 ( 5 + 2 ) 17 ( 17 + 1 ) + 5 ( 5 − 2 ) 85 ( 85 − 2 ) 5 ( 5 + 2 ) 85 ( 85 + 2 ) + 17 ( 17 − 1 ) 85 ( 85 − 2 ) 17 ( 17 + 1 ) 85 ( 85 + 2 ) = ( 5 − 2 ) ( 17 − 1 ) ( 5 + 2 ) ( 17 + 1 ) + ( 5 − 2 ) ( 85 − 2 ) ( 5 + 2 ) ( 85 + 2 ) + ( 17 − 1 ) ( 85 − 2 ) ( 17 + 1 ) ( 85 + 2 ) = ( 5 − 2 ) ( 5 − 2 ) ( 17 − 1 ) ( 17 − 1 ) ( 5 + 2 ) ( 5 − 2 ) ( 17 + 1 ) ( 17 − 1 ) + ( 5 − 2 ) ( 5 − 2 ) ( 85 − 2 ) ( 85 − 2 ) ( 5 + 2 ) ( 5 − 2 ) ( 85 + 2 ) ( 85 − 2 ) + ( 17 − 1 ) ( 17 − 1 ) ( 85 − 2 ) ( 85 − 2 ) ( 17 + 1 ) ( 17 − 1 ) ( 85 + 2 ) ( 85 − 2 ) = ( 5 − 2 ) 2 ( 17 − 1 ) 2 ( 1 ) ( 16 ) + ( 5 − 2 ) 2 ( 85 − 2 ) 2 ( 1 ) ( 81 ) + ( 17 − 1 ) 2 ( 85 − 2 ) 2 ( 16 ) ( 81 ) = ( 5 − 2 ) ( 17 − 1 ) ( 1 ) ( 4 ) + ( 5 − 2 ) ( 85 − 2 ) ( 1 ) ( 9 ) + ( 17 − 1 ) ( 85 − 2 ) ( 4 ) ( 9 ) = 9 ( 85 − 2 17 − 5 + 2 ) + 4 ( 5 17 − 2 85 − 2 5 + 4 ) + 17 5 − 85 − 2 17 + 2 36 = 36 36 = 1 \small \sqrt{\dfrac{(5-2\sqrt{5})(17-\sqrt{17})}{(5+2\sqrt{5})(17+\sqrt{17})}} + \sqrt{\dfrac{(5-2\sqrt{5})(85-2\sqrt{85})}{(5+2\sqrt{5})(85+2\sqrt{85})}} + \sqrt{\dfrac{(17-\sqrt{17})(85-2\sqrt{85})}{(17+\sqrt{17})(85+2\sqrt{85})}} \\ \small = \sqrt{\dfrac{\sqrt{5}(\sqrt{5}-2)\sqrt{17}(\sqrt{17}-1)}{\sqrt{5}(\sqrt{5}+2)\sqrt{17}(\sqrt{17}+1)}} + \sqrt{\dfrac{\sqrt{5}(\sqrt{5}-2) \sqrt{85}(\sqrt{85}-2)}{\sqrt{5}(\sqrt{5}+2)\sqrt{85}(\sqrt{85}+2)}} + \sqrt{\dfrac{\sqrt{17}(\sqrt{17}-1) \sqrt{85}(\sqrt{85}-2)}{\sqrt{17}(\sqrt{17}+1)\sqrt{85}(\sqrt{85}+2)}} \\ \small = \sqrt{\dfrac{(\sqrt{5}-2)(\sqrt{17}-1)}{(\sqrt{5}+2)(\sqrt{17}+1)}} + \sqrt{\dfrac{(\sqrt{5}-2)(\sqrt{85}-2)}{(\sqrt{5}+2)(\sqrt{85}+2)}} + \sqrt{\dfrac{(\sqrt{17}-1)(\sqrt{85}-2)}{(\sqrt{17}+1)(\sqrt{85}+2)}} \\ \small = \sqrt{\dfrac{(\sqrt{5}-2)(\sqrt{5}-2)(\sqrt{17}-1)(\sqrt{17}-1)}{(\sqrt{5}+2)(\sqrt{5}-2)(\sqrt{17}+1)(\sqrt{17}-1)}} + \sqrt{\dfrac{(\sqrt{5}-2)(\sqrt{5}-2)(\sqrt{85}-2)(\sqrt{85}-2)}{(\sqrt{5}+2)(\sqrt{5}-2)(\sqrt{85}+2)(\sqrt{85}-2)}} \\ \small \quad \quad + \sqrt{\dfrac{(\sqrt{17}-1)(\sqrt{17}-1)(\sqrt{85}-2)(\sqrt{85}-2)}{(\sqrt{17}+1)(\sqrt{17}-1)(\sqrt{85}+2)(\sqrt{85}-2)}} \\ \small = \sqrt{\dfrac{(\sqrt{5}-2)^2(\sqrt{17}-1)^2}{(1)(16)}} + \sqrt{\dfrac{(\sqrt{5}-2)^2(\sqrt{85}-2)^2}{(1)(81)}} + \sqrt{\dfrac{(\sqrt{17}-1)^2(\sqrt{85}-2)^2}{(16)(81)}} \\ \small = \dfrac{(\sqrt{5}-2)(\sqrt{17}-1)}{(1)(4)} + \dfrac{(\sqrt{5}-2)(\sqrt{85}-2)}{(1)(9)} + \dfrac{(\sqrt{17}-1)(\sqrt{85}-2)}{(4)(9)} \\ \small = \dfrac{9(\sqrt{85}-2\sqrt{17}-\sqrt{5}+2) + 4(5\sqrt{17}-2\sqrt{85}-2\sqrt{5}+4) + 17\sqrt{5}-\sqrt{85}-2\sqrt{17} +2}{36} \\ \small = \dfrac{36}{36} = \boxed{1}

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Can i ask you went from the second to third line of working?

Wanyu Tang - 5 years, 8 months ago

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I think you mean line 3 to line 4. I have added two lines. Hope they are useful.

Chew-Seong Cheong - 5 years, 8 months ago

Excellent solution sir. Did the same :)

John Frank - 5 years, 4 months ago

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