Solve the System: Equalizing |x+2| and |x-2| with x⁴ = 0

$\begin{cases} |x+2|=|x-2| \\ x^4=0 \end{cases}$

To solve this problem, we'll follow these steps:

• Step 1: Solve the polynomial equation $x^4 = 0$.
• Step 2: Verify the solution against the absolute value equation $|x+2| = |x-2|$.
• Step 3: Ensure the solution matches a provided choice, if necessary.

Now, let's work through each step:

Step 1: Solving $x^4 = 0$ yields $x = 0$, as any non-zero values to the power of 4 would not equal zero.

Step 2: Substituting $x = 0$ into the absolute value equation:

$|0+2| = |0-2|$

$|2| = |-2|$

$2 = 2$, thus the condition is satisfied.

Step 3: Compare with provided answer choices. Since the correct choice must satisfy both equations, $x = 0$ is consistent with option (1).

Therefore, the solution to the problem is $x = 0$.