<s>[INST] Find the smallest possible value for: $a+b$ If $a,b$ are positive integers with $a, b > 1$, and
$$\sqrt{a\sqrt{a\sqrt{a}}}=b,$$
find the smallest possible value for $a+b$.
 [/INST] We have: $a^7 = b^8 \to a = b\cdot b^{\frac{1}{7}}$. Take $c = b^{\frac{1}{7}} \to a = c^7\cdot c = c^8, b = c^7$. Thus: $a+b = c^7+c^8$. Clearly the minimum occurs if $c = 2$, for $a = 2^8 = 256, b = 2^7 = 128$.
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