1. **State the problem:** We are given two equations: $$ (x + y)^6 = 64 $$ $$ (x - y)^2 = 16 $$ We need to find the value of $x + y$. 2. **Analyze the equations:** From the first equation, we have a sixth power of $x + y$ equal to 64. From the second equation, the square of $x - y$ equals 16. 3. **Solve the second equation first:** $$ (x - y)^2 = 16 $$ Taking the square root of both sides: $$ x - y = \pm 4 $$ 4. **Solve the first equation:** $$ (x + y)^6 = 64 $$ Rewrite 64 as a power of 2: $$ 64 = 2^6 $$ So: $$ (x + y)^6 = 2^6 $$ Taking the sixth root of both sides: $$ x + y = \pm 2 $$ 5. **Conclusion:** The possible values for $x + y$ are 2 or -2. Since the problem asks for $x + y$, the answer is: $$ \boxed{\pm 2} $$