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 $N^{\frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}}=N^{frac{25}{36}}$ 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