The term simply reduces to (2/\sqrt{5}+1)^{24} after you cancel out the sqrt{6} terms. They really then want you to recognise that this is the inverse of the "Golden Ratio" G=(\sqrt{5}+1)/2 . Its inverse is G^{-1}=(\sqrt{5}-1)/2 so the answer is is G^{-24} ~(1.618)^{24}) which is a very small number. Using a calculator to directly evaluate the initial expression should give a match. The Golden ratio is also a solution of the quadratic G^{2}-G-1=0. You really solved the problem at around 2.00 if you recognise the Golden Ratio, although many otherwise able students may never have encountered it.
√24/(√30+√6)=2/(√5+1)=1/Φ=φ, where Φ=(√5+1)/2 is the Golden Ratio and φ=2/(√5+1)=(√5-1)/2 is the inverse Golden Ratio. Fibonacci showed that Φ^24=46368Φ+28657=103682.... He also showed that φ^24=28657-46368φ=0.00000964488.... Also φ^24=28657+23184-23184√5=51841-23184√5
Nice olympiad problem
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very nice explaination
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The term simply reduces to (2/\sqrt{5}+1)^{24} after you cancel out the sqrt{6} terms. They really then want you to recognise that this is the inverse of the "Golden Ratio" G=(\sqrt{5}+1)/2 . Its inverse is G^{-1}=(\sqrt{5}-1)/2 so the answer is is G^{-24} ~(1.618)^{24}) which is a very small number. Using a calculator to directly evaluate the initial expression should give a match. The Golden ratio is also a solution of the quadratic G^{2}-G-1=0. You really solved the problem at around 2.00 if you recognise the Golden Ratio, although many otherwise able students may never have encountered it.
Perfect example to watch here
Correct solution
right solution
√24/(√30+√6)=2/(√5+1)=1/Φ=φ, where Φ=(√5+1)/2 is the Golden Ratio and φ=2/(√5+1)=(√5-1)/2 is the inverse Golden Ratio.
Fibonacci showed that Φ^24=46368Φ+28657=103682.... He also showed that φ^24=28657-46368φ=0.00000964488....
Also φ^24=28657+23184-23184√5=51841-23184√5
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