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

To find the smallest number, we convert them to a common base or a common exponent for easier comparison.

1. Convert numbers with base 2:

   • 2^40
   • 4^18 = (2^2)^18 = 2^(2*18) = 2^36
   • 8^12 = (2^3)^12 = 2^(3*12) = 2^36

2. Now the numbers are: 2^40, 3^21, 2^36, 2^36.

3. Comparing 2^40, 2^36, and 2^36, it's clear that 2^36 is the smallest among these three.

4. So, the comparison boils down to finding the smaller between 3^21 and 2^36. To compare 3^21 and 2^36, we can find a common factor for their exponents (21 and 36). The greatest common divisor of 21 and 36 is 3.

   • 3^21 = (3^7)^3
   • 2^36 = (2^12)^3

5. Now we compare the bases: 3^7 and 2^12.

   • 3^7 = 3 * 3 * 3 * 3 * 3 * 3 * 3 = 9 * 9 * 9 * 3 = 81 * 27 = 2187
   • 2^12 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4096 (since 2^10 = 1024, 2^12 = 1024 * 4 = 4096)

6. Since 2187 < 4096, it means 3^7 < 2^12. Therefore, (3^7)^3 < (2^12)^3, which implies 3^21 < 2^36.

7. Since 3^21 is smaller than 2^36, and 2^36 was the smallest among the base-2 numbers, 3^21 is the overall smallest number.

The final answer is B) 3^21.