https://file.aiquickdraw.com/mathbot/file/1726299655189wvtk4rjr.jpg

The image contains two equations:

\[
(x + y)^6 = 64
\]
\[
(x - y)^2 = 16
\]
We are tasked with finding \( x + y \).

### Step-by-step solution:

1. From the first equation, we know that:
   \[
   (x + y)^6 = 64
   \]
   Taking the 6th root of both sides:
   \[
   x + y = 64^{1/6} = 2
   \]
   
2. Now that we know \( x + y = 2 \), we can substitute this into the second equation:
   \[
   (x - y)^2 = 16
   \]
   Taking the square root of both sides:
   \[
   x - y = \pm 4
   \]
   
3. We now have two cases to consider based on \( x - y \):
   
   **Case 1:**
   \[
   x - y = 4
   \]
   We can solve this system:
   \[
   x + y = 2
   \]
   \[
   x - y = 4
   \]
   Adding these two equations:
   \[
   2x = 6 \quad \Rightarrow \quad x = 3
   \]
   Substituting \( x = 3 \) into \( x + y = 2 \):
   \[
   3 + y = 2 \quad \Rightarrow \quad y = -1
   \]

   **Case 2:**
   \[
   x - y = -4
   \]
   Solving this system:
   \[
   x + y = 2
   \]
   \[
   x - y = -4
   \]
   Adding these two equations:
   \[
   2x = -2 \quad \Rightarrow \quad x = -1
   \]
   Substituting \( x = -1 \) into \( x + y = 2 \):
   \[
   -1 + y = 2 \quad \Rightarrow \quad y = 3
   \]

Thus, we have two possible solutions:
1. \( x = 3, y = -1 \)
2. \( x = -1, y = 3 \)

In both cases, \( x + y = 2 \).

### Final answer:
\[
x + y = 2
\]

Would you like further clarification or additional details?

Here are 5 related questions:
1. What is the significance of taking the 6th root in the first equation?
2. How do we decide when to consider positive and negative square roots?
3. Could this problem have had more than two solutions?
4. How does solving systems of equations help with simultaneous equations?
5. Can we use a graphical method to solve this system?

**Tip:** When solving powers of variables, always consider both positive and negative root possibilities.

(x + y)^6 = 64 and (x - y)^2 = 16. Find x + y.

This step-by-step solution explains how to find x + y by solving two equations: (x + y)^6 = 64 and (x - y)^2 = 16. Learn how to use exponentiation, square roots, and systems of equations. Suitable for students in Grades 9-12.

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