Sure! Let's go through the problem step-by-step.
### Step 1: Calculate [tex]\((1 + i)^4\)[/tex]
First, we want to calculate [tex]\((1 + i)^4\)[/tex]. Let's work through the expansion using the binomial theorem or any other convenient method:
[tex]\[
(1 + i)^4
\][/tex]
Using binomial expansion:
[tex]\[
(1 + i)^4 = \sum_{k=0}^{4} \binom{4}{k} 1^{4-k} i^k
\][/tex]
Writing out all terms:
[tex]\[
\binom{4}{0} \cdot 1^4 \cdot i^0 + \binom{4}{1} \cdot 1^3 \cdot i^1 + \binom{4}{2} \cdot 1^2 \cdot i^2 + \binom{4}{3} \cdot 1^1 \cdot i^3 + \binom{4}{4} \cdot 1^0 \cdot i^4
\][/tex]
Simplifying the coefficients:
[tex]\[
1 \cdot 1 + 4 \cdot i + 6 \cdot (-1) + 4 \cdot (-i) + 1 \cdot 1
\][/tex]
This simplifies to:
[tex]\[
1 + 4i - 6 - 4i + 1
\][/tex]
Combining like terms:
[tex]\[
(1 + 1 - 6) + (4i - 4i) = -4
\][/tex]
Thus, we have:
[tex]\[
(1 + i)^4 = -4
\][/tex]
### Step 2: Calculate [tex]\(\left(1 + \frac{1}{i}\right)^4\)[/tex]
Next, let's calculate [tex]\(\left(1 + \frac{1}{i}\right)^4\)[/tex]. First, we need to simplify [tex]\(1 + \frac{1}{i}\)[/tex]:
[tex]\[
1 + \frac{1}{i} = 1 - i \quad \text{(since } \frac{1}{i} = -i\text{)}
\][/tex]
So, we need to find:
[tex]\[
(1 - i)^4
\][/tex]
Using the same method as before, we expand:
[tex]\[
(1 - i)^4 = \sum_{k=0}^{4} \binom{4}{k} 1^{4-k} (-i)^k
\][/tex]
Writing out all terms:
[tex]\[
\binom{4}{0} \cdot 1^4 \cdot (-i)^0 + \binom{4}{1} \cdot 1^3 \cdot (-i)^1 + \binom{4}{2} \cdot 1^2 \cdot (-i)^2 + \binom{4}{3} \cdot 1^1 \cdot (-i)^3 + \binom{4}{4} \cdot 1^0 \cdot (-i)^4
\][/tex]
Simplifying the coefficients and powers:
[tex]\[
1 \cdot 1 - 4 \cdot i + 6 \cdot (-1) - 4 \cdot (-i) + 1
\][/tex]
This simplifies to:
[tex]\[
1 - 4i - 6 + 4i + 1
\][/tex]
Combining like terms:
[tex]\[
(1 + 1 - 6) + (-4i + 4i) = -4
\][/tex]
Thus, we have:
[tex]\[
\left(1 + \frac{1}{i}\right)^4 = -4
\][/tex]
### Step 3: Combine the Results
Finally, combine the results of the two expressions:
[tex]\[
(1 + i)^4 \cdot \left(1 + \frac{1}{i}\right)^4 = (-4) \cdot (-4)
\][/tex]
Solving this:
[tex]\[
(-4) \cdot (-4) = 16
\][/tex]
Therefore, the proof is complete:
[tex]\[
(1 + i)^4 \cdot \left(1 + \frac{1}{i}\right)^4 = 16
\][/tex]
