Arrange in increasing order: $2^{30},\ 3^{20},\ 6^{10}$.

Arrange in increasing order: $2^{30},\ 3^{20},\ 6^{10}$.
  1. $2^{30} < 3^{20} < 6^{10}$
  2. $6^{10} < 2^{30} < 3^{20}$
  3. $3^{20} < 6^{10} < 2^{30}$
  4. $2^{30} < 6^{10} < 3^{20}$

Solution

Common index 10: $2^{30} = 8^{10}$, $3^{20} = 9^{10}$, $6^{10} = 6^{10}$. Since $6 < 8 < 9$, the order is $6^{10} < 2^{30} < 3^{20}$.

Asked in: IMO

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