This question was uploaded on 28/08/21 on social media accounts.
This question was asked during AIME (American Invitational Mathematics Examination). During 2005 this was asked as Problem Number 7.
Given equation:
\[\left(\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}+1\right)^{48}\]
Step 1:
Multiply Numerator and Denominator with:
\[\sqrt[16]{5}-1\]
Now equation becomes:
\[\left(\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}\frac{\color{blue}{(\sqrt[16]{5}-1)}}{\color{blue}{(\sqrt[16]{5}-1)}}+1\right)^{48}\]
Step 2:
At Denominator:
\[(\sqrt[16]{5}+1)(\sqrt[16]{5}-1)=\sqrt[8]{5}-1\]
Now equation becomes:
\[\left(\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)}\frac{\color{blue}{(\sqrt[16]{5}-1)}}{\color{red}{(\sqrt[8]{5}-1)}}+1\right)^{48}\]
Similarly,
\[(\sqrt[8]{5}+1)(\sqrt[8]{5}-1)=\sqrt[4]{5}-1\]
\[(\sqrt[4]{5}+1)(\sqrt[4]{5}-1)=\sqrt{5}-1\]
\[(\sqrt{5}+1)(\sqrt{5}-1)=5-1=4\]
Equation shrinks as follows:
\[\left(\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)}\frac{\color{blue}{(\sqrt[16]{5}-1)}}{\color{orange}{(\sqrt[4]{5}-1)}}+1\right)^{48}\]
\[=\left(\frac{4}{(\sqrt{5}+1)}\frac{\color{blue}{(\sqrt[16]{5}-1)}}{\color{fuchsia}{(\sqrt{5}-1)}}+1\right)^{48}\]
\[=\left(\frac{4{\color{blue}{(\sqrt[16]{5}-1)}}}{\color{green}{4}}+1\right)^{48}\]
\[=\left(\sqrt[16]{5}-1+1\right)^{48}\]
\[=(\sqrt[16]{5})^{48}=5^3=125 \]
Puzzle related to Algebra, Equation.
