https://file.aiquickdraw.com/mathbot/file/1725821522698hzn8553g.jpg

The image contains two equations:

\[
(x + y)^6 = 64
\]
\[
(x - y)^2 = 16
\]

We are asked to find the value of \(x + y\).

### Step 1: Solve the first equation
\[
(x + y)^6 = 64
\]
Taking the sixth root of both sides:
\[
x + y = \sqrt[6]{64}
\]
Since \(64 = 2^6\), we have:
\[
x + y = 2
\]

### Step 2: Solve the second equation
\[
(x - y)^2 = 16
\]
Taking the square root of both sides:
\[
x - y = \sqrt{16}
\]
This gives two possible values:
\[
x - y = 4 \quad \text{or} \quad x - y = -4
\]

### Step 3: Combine the results
Now, we have two scenarios based on the second equation.

#### Case 1: \(x - y = 4\)
Adding the two equations:
\[
(x + y) + (x - y) = 2 + 4 = 6
\]
This simplifies to:
\[
2x = 6 \quad \Rightarrow \quad x = 3
\]
Substituting \(x = 3\) into \(x + y = 2\):
\[
3 + y = 2 \quad \Rightarrow \quad y = -1
\]
So, in this case, \(x = 3\) and \(y = -1\).

#### Case 2: \(x - y = -4\)
Adding the two equations:
\[
(x + y) + (x - y) = 2 - 4 = -2
\]
This simplifies to:
\[
2x = -2 \quad \Rightarrow \quad x = -1
\]
Substituting \(x = -1\) into \(x + y = 2\):
\[
-1 + y = 2 \quad \Rightarrow \quad y = 3
\]
So, in this case, \(x = -1\) and \(y = 3\).

### Conclusion
The value of \(x + y\) is \(2\), and we've found the two possible pairs for \(x\) and \(y\): \( (3, -1) \) and \( (-1, 3) \).

Would you like more details on any specific part of the solution?

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**Related Questions:**
1. What are the general properties of even and odd powers when solving such equations?
2. How can we solve similar equations involving different roots, like cube roots or fourth roots?
3. What other methods can be used to solve systems of nonlinear equations like this one?
4. How does knowing the symmetry of an equation help in solving it?
5. How would the solution change if one of the powers in the equations was different, say \((x+y)^4\)?

**Tip:** Always check all possible cases when solving equations involving roots, as they can produce multiple valid solutions.

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

This problem involves solving a system of nonlinear equations with powers and roots. Learn the step-by-step process to find x + y using algebraic methods. Suitable for students in Grades 9-12.

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