If M varies directly as N and inversely as the root of P. Given that M = 3, N = 5 and P = 25. Find the value of P when M = 2 and N = 6.

The problem tells us that "M varies directly as N and inversely as the root of P". This means that M is directly proportional to N and inversely proportional to the square root of P, which can be written mathematically as: M = k(N / sqrt(P)) where k is a constant of proportionality. We can solve for k by using the given values of M, N, and P. When M = 3, N = 5, and P = 25, we can plug these values into the equation above and solve for k: 3 = k(5 / sqrt(25)) 3 = k(5 / 5) 3 = k So, we have found that k = 3. Now, we can use this value of k to find P when M = 2 and N = 6. We can rearrange the equation above to solve for P: M = k(N / sqrt(P)) sqrt(P) = k(N / M) P = (k(N / M))^2 Plugging in k = 3, N = 6, and M = 2, we get: P = (3(6 / 2))^2 P = 9(3)^2 P = 81 Therefore, the value of P when M = 2 and N = 6 is 81. So, option D is the correct answer.