Calculating \((1+i)^4 + (1-i)^4\)

This problem involves calculating the sum of powers of complex numbers. We need to find the value of the expression \((1+i)^4 + (1-i)^4\), where the imaginary unit \(i\) is defined as \(\sqrt{-1}\).

Evaluating \((1+i)^4\)

We can evaluate \((1+i)^4\) in a couple of ways. One common method is to first calculate \((1+i)^2\).

• \((1+i)^2 = 1^2 + 2(1)(i) + i^2\)
• Since \(i^2 = -1\), this becomes \(1 + 2i - 1\).
• Simplifying, we get \((1+i)^2 = 2i\).

Now, we can find \((1+i)^4\) by squaring the result of \((1+i)^2\):

• \((1+i)^4 = ((1+i)^2)^2\)
• Substituting the value we found: \((1+i)^4 = (2i)^2\).
• Squaring this gives \(4i^2\).
• Again, since \(i^2 = -1\), we have \(4(-1)\).
• Therefore, \((1+i)^4 = -4\).

Evaluating \((1-i)^4\)

Similarly, we evaluate \((1-i)^4\). First, let's find \((1-i)^2\).

• \((1-i)^2 = 1^2 - 2(1)(i) + i^2\)
• Substituting \(i^2 = -1\): \(1 - 2i - 1\).
• Simplifying, we get \((1-i)^2 = -2i\).

Now, we find \((1-i)^4\) by squaring \((1-i)^2\):

• \((1-i)^4 = ((1-i)^2)^2\)
• Substituting the value we found: \((1-i)^4 = (-2i)^2\).
• Squaring this gives \(4i^2\).
• Since \(i^2 = -1\), this is \(4(-1)\).
• Therefore, \((1-i)^4 = -4\).

Summing the Results

Finally, we need to add the results of \((1+i)^4\) and \((1-i)^4\) together.

• The expression is \((1+i)^4 + (1-i)^4\).
• Substituting the values we calculated: \((-4) + (-4)\).
• The sum is \(-8\).

Conclusion

The value of \((1+i)^4 + (1-i)^4\) is \(-8\). This corresponds to the option provided.