Which is greater: $2^{30}$ or $3^{20}$?

Which is greater: $2^{30}$ or $3^{20}$?
  1. $2^{30}$
  2. $3^{20}$
  3. Equal
  4. Cannot be compared

Solution

Bring to common index 10: $2^{30} = (2^{3})^{10} = 8^{10}$ and $3^{20} = (3^{2})^{10} = 9^{10}$. Since $9 > 8$, $3^{20} > 2^{30}$.

Asked in: IMO

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