Indonesia Open Mathematical Contest

Relevant wiki: Simplifying Expressions with Radicals - Intermediate

$\begin{aligned} X & = \sqrt [4]{2^4+2^{11}+2^{15}+2^{16}+2^{21}+2^{24}} \\ & = 2\sqrt [4]{\color{#3D99F6}{1+2^{7}+2^{11}+2^{12}+2^{17}+2^{20}}} \quad \quad \small \color{#3D99F6}{\text{Assuming the form of }(1+x)^4 \implies x^4 = 2^{20} \implies x = 2^5 =32} \\ & = 2\sqrt [4]{\color{#3D99F6}{1+4(2^{5})+(2+4)(2^{5})^2+4(2^{5})^3+(2^{5})^4}} \\ & = 2\sqrt [4]{\color{#3D99F6}{1+4(2^{5})+6(2^{5})^2+4(2^{5})^3+(2^{5})^4}} \quad \quad \small \color{#3D99F6}{\text{Note that }(1+x)^4=1+4x+6x^2+4x^3+x^4} \\ & = 2\sqrt [4]{(1+2^5)^4} = 2(1+2^5) = 2(1+32) = \boxed{66} \end{aligned}$

Lets simplify the part inside the radical first:

$2^{4}+2^{11}+2^{15}+2^{16}+2^{21}+2^{24}$

$=2^{4}(1+2^{7}+2^{11}+2^{12}+2^{17}+2^{20})$

$=2^{4}[1+2^{7}+2^{11}+2^{17}+(2^{12}+2^{20})]$

$=2^{4}[1+2^{7}+2^{11}+2^{17}+(2^{6}+2^{10})^{2}-2^{17}]$

$=2^{4}[1^{2}+(2^{6}+2^{10})^{2}+2^{7}+2^{11}]$

$=2^{4}[(1+2^{6}+2^{10})^{2}-2^{7}-2^{11}+2^{7}+2^{11}]$

$=2^{4}[1^{2}+2^{5}+(2 \times 1 \times 2^{5})]^{2}$

$=2^{4}[1+2^{5}]^{4}=(2 \times 33)^{4}=66^{4}$

Now, putting it under the radical, we have $66$ as the final answer.

For $\sqrt[4]{2^4+2^{11}+2^{15}+2^{16}+2^{21}+2^{24}}$

Here's the steps:

$\begin{aligned} \sqrt[4]{ \color{#D61F06}{2^{4}} + \color{#EC7300}{2^{11}} + \color{#CEBB00}{2^{15}} + \color{#20A900}{2^{16}}+ \color{#3D99F6}{2^{21}}+\color{#302B94}{2^{24}}} &= \sqrt[4]{ \color{#D61F06}{16}+ \color{#EC7300}{2048}+\color{#CEBB00}{32768}+\color{#20A900}{65536}+\color{#3D99F6}{2097152}+\color{#302B94}{16777216} } \\&= \sqrt[4]{18974736} \\&= 66 \space \square \end{aligned}$

$\LARGE \text{ADIOS!!!}$