2020 AMC 12A Problems/Problem 13

Problem

There are integers $a, b,$ and $c,$ each greater than $1,$ such that

$\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}$

$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$

Solution

$\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}}$ can be simplified to $N^{\frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}}.

The equation is then$ (Error compiling LaTeX. Unknown error_msg)N^{\frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}}=N^{frac{25}{36}} $which implies that$ \frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}=\frac{25}{36}. $a$ has to be $2$ since $\frac{25}{36}>\frac{1}{2}$ . $b$ being $3$ will make the fraction $frac{2}{3}$ which is close to $frac{25}{36}$ . Finally, with $c$ being $6$ , the fraction becomes $frac{25}{36}$ . In this case $a, b,$ and $c$ work, which means that $b$ must equal $\boxed{\textbf{(B) } 3.}$ ~lopkiloinm

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2020 AMC 12A Problems/Problem 13

Problem

Solution