If x and y are real numbers such that `16^(x^(2)+y)+16^(x+y^2)=1`, then find the values of x and y.

Explanation

To solve the equation \( 16^{(x^2 + y)} + 16^{(x + y^2)} = 1 \), we can follow these steps:

### Step 1: Analyze the equation
We know that \( 16^{(x^2 + y)} \) and \( 16^{(x + y^2)} \) are both positive since any power of a positive number is positive. Therefore, the sum of two positive numbers can only equal 1 if both terms are less than or equal to 1.

### Step 2: Apply the AM-GM inequality
Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we can write:
\[
...