5. Let $a$ and $b$ be natural numbers greater than 1, for which $\sqrt{a \sqrt{a \sqrt{a}}}=b$. What is the smallest possible value of the sum $a+b$?

Solve the problem independently. You have 150 minutes for solving. The use of notes, literature, or a pocket calculator is not allowed.

Mathematical Competition for High School Students in Slovenia

## Invitational Competition

March 28, 2003

## Problems for 2nd Year
I/5. From $\sqrt{a \sqrt{a \sqrt{a}}}=b$ we sequentially get $a \sqrt{a \sqrt{a}}=b^{2}, a^{3} \sqrt{a}=b^{4}$ and $a^{7}=b^{8}$. This means that the number $a$ is an 8th power, and the number $b$ is a 7th power of a natural number greater than 1. Since we are looking for the smallest possible sum of the numbers $a$ and $b$, we take $a=2^{8}$ and $b=2^{7}$. Then $a+b=256+128=384$.

Reformulation to $a^{7}=b^{8}: 2$ points. Conclusion that $a$ is an 8th power, and $b$ is a 7th power of a natural number: $\mathbf{2}$ points. Conclusion that $a=2^{8}$ and $b=2^{7}$ due to minimality: $\mathbf{2}$ points. Answer $a+b=384: 1$ point.