The two positive integer solutions of the equation x^2 - mx + n = 0 are k and t, where m and n are both prime numbers and k > t. What is the value of m^n + n^m + k^t + t^k
It's 20
To find the value of m^n + n^m + k^t + t^k, we first need to find the values of m, n, k, and t. Let's break down the problem step by step:
Step 1: Solve the quadratic equation x^2 - mx + n = 0 to find the values of k and t.
Since the coefficient of x^2 is 1, the sum of the roots of the quadratic equation is equal to the negative coefficient of x, which is -m. So, k + t = m.
The product of the roots of the quadratic equation is equal to the constant term, which is n. So, k * t = n.
Step 2: Identify the prime numbers m and n.
Since both m and n are prime numbers, they can only be divided by 1 and themselves. Start by substituting some possible prime numbers for m until you find a set of values for k and t that satisfy the given conditions.
For each prime number you choose for m, calculate k + t and k * t. If k + t is equal to that prime number and k * t is equal to another prime number, you have found your solution.
Note that k > t, so consider the prime numbers in increasing order.
Step 3: Once you have found the values of m, n, k, and t, calculate m^n, n^m, k^t, and t^k.
Raise each value to the power of the given exponent and calculate their individual results.
Step 4: Add the results from step 3 to find the final value.
Add m^n, n^m, k^t, and t^k to get the final value.
Continue this process until you find a consistent set of values for m, n, k, and t and then calculate the final value.