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

• Convert all numbers to the same base, which is 2, to facilitate comparison.

• 2⁴⁰ remains 2⁴⁰.

• 3²¹ cannot be converted to base 2 directly, so compare it with the others by estimating or converting others to base 3 or using logarithms; however, a simpler method is to convert the base-2 numbers to a common exponent or compare magnitudes.

• Let's convert 4¹⁸ and 8¹² to base 2: 4¹⁸ = (2²)¹⁸ = 2³⁶ and 8¹² = (2³)¹² = 2³⁶.

• So, 4¹⁸ and 8¹² are equal to 2³⁶.

• Now compare 2⁴⁰ and 2³⁶. Since 40 > 36, 2⁴⁰ > 2³⁶.

• Now compare 3²¹ with 2³⁶. Note that 3²¹ = (3³)⁷ = 27⁷ and 2³⁶ = (2⁶)⁶ = 64⁶. This is not direct. Let's use logarithms or simple bounds. 3²¹ ≈ 1.04 × 10¹⁰. 2³⁶ ≈ 6.87 × 10¹⁰. 2⁴⁰ ≈ 1.09 × 10¹².

• Therefore, 3²¹ (approx 10¹⁰) is smaller than 2³⁶ (approx 10¹⁰ but larger coefficient) and 2⁴⁰ (approx 10¹²). Specifically, 3²¹ < 2³⁶ < 2⁴⁰.

• Thus, 3²¹ is the smallest number.