Which number amongst 2^40, 3^21, 4^18 and 8^12 is the smallest?

• (a) (2^{40}): Keep as base 2. Value is (2^{40}). Not the smallest.
• (b) (3^{21}): Compare by size using logs/powers: (3^{21} < 2^{40}) since (21\log 3 < 40\log 2). Smallest.
• (c) (4^{18}): (4^{18}=(2^2)^{18}=2^{36}). Since (2^{36} < 2^{40}), it is smaller than (a), but compare with (b): (2^{36} > 3^{21}). Not the smallest.
• (d) (8^{12}): (8^{12}=(2^3)^{12}=2^{36}), same as (4^{18}). Not the smallest.

Therefore, the smallest number is (3^{21}).