Which number amongst 2⁴⁰, 3²¹, 4¹⁸ and 8¹² is the smallest?

Correct Option (B)

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.

Incorrect 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.