Question

Find the value of k, if k is the smallest positive integer for which k ^k^k >k^11k is true.

Answer 1

To find the value of k, we need to solve the inequality k^(k^k) > k^(11k). The smallest positive integer value for k is 12.

Step-by-step explanation:

To find the value of k, we need to solve the inequality k^(k^k) > k^(11k).

First, let's simplify the exponents on both sides of the inequality:

k^(k^k) = k^(k*k) = k^(k²)

k^(11k) = k^11k

Now we can set up the inequality:

k^(k²) > k^11k

Since k is a positive integer, the inequality can be simplified further:

k² > 11k

Divide both sides of the inequality by k:

k > 11

Therefore, the smallest positive integer value satisfying this inequality for k is 12, fulfilling the condition where k is greater than 11, showcasing how to determine the solution for k in this context.