| Dictionary definition | Usage | |
|---|---|---|
| A. To extend outside of or enlarge beyond, used chiefly in strictly physical relations | E | The mercy of God exceeds our finite minds |
| B. To be greater than or superior to | F | Their accomplishments exceeded our expectation. |
| C. Be beyond the comprehension of | G | He exceeded his authority when he paid his brother’s gambling debts with money from the trust. |
| D. To go beyond a limit set by (as an authority or privilege) | H | If this rain keeps up, the river will exceed its banks by morning. |


For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: