To determine the smallest number among 2⁴⁰, 3²¹, 4¹⁸, and 8¹², we standardize their bases for comparison.

First, convert 4¹⁸ and 8¹² to base 2:
- 4¹⁸ = (2²)¹⁸ = 2³⁶
- 8¹² = (2³)¹² = 2³⁶
Thus, 4¹⁸ = 8¹² = 2³⁶.
Next, compare 2⁴⁰ with 2³⁶. Since 2⁴⁰ > 2³⁶, 2⁴⁰ is not the smallest.
The comparison now narrows down to 3²¹ and 2³⁶. To compare these, we can find a common exponent or compare simplified powers:
- Consider 3⁷ and 2¹².
- Calculate their values: 3⁷ = 2187 and 2¹² = 4096.
- Since 3⁷ < 2¹², raising both sides to the power of 3 preserves the inequality: (3⁷)³ < (2¹²)³.
- This simplifies to 3²¹ < 2³⁶.

Therefore, 3²¹ is the smallest number among the given options.

2⁴⁰: This value is greater than 2³⁶, which is equal to 4¹⁸ and 8¹². Hence, it is not the smallest.
4¹⁸: This value is equal to 2³⁶. As established, 3²¹ is smaller than 2³⁶. Therefore, 4¹⁸ is not the smallest.
8¹²: This value is also equal to 2³⁶. Since 3²¹ is smaller than 2³⁶, 8¹² is not the smallest.