Final answer:
To determine the insect population after one year given that it doubles every month, an exponential expression N = 400 * 2¹² is used, yielding 1,638,400 insects after one year of growth.
Step-by-step explanation:
The number of insects in a colony doubling every month can be represented by an exponential expression. The exponential growth formula is N = N0 * 2ⁿ, where N is the final population size, N0 is the initial population size, and n is the number of time periods (months) of growth. For the colony of insects with an initial population of 400 after one year (which is 12 months), the formula becomes N = 400 * 2¹².
To solve for N, we perform the calculation: N = 400 * 2¹² = 400 * 4096 = 1,638,400. Therefore, after one year, the insect colony will have grown to 1,638,400 insects.