Let's simplify the given expression:

\[
\frac{(1+i)^7 \cdot (1-i\sqrt{3})^3}{(1-i)^7 \cdot (1+i\sqrt{3})}
\]

### Step 1: Simplify \((1+i)^7\) and \((1-i)^7\)
The expressions \((1+i)\) and \((1-i)\) are complex numbers. Their magnitudes are the same:

\[
|1+i| = |1-i| = \sqrt{1^2 + 1^2} = \sqrt{2}
\]

The argument (angle) of \(1+i\) is \(\frac{\pi}{4}\) and for \(1-i\), it is \(-\frac{\pi}{4}\).

Thus:

\[
(1+i)^7 = (\sqrt{2})^7 \left(\cos\left(\frac{7\pi}{4}\right) + i\sin\left(\frac{7\pi}{4}\right)\right)
\]
\[
(1-i)^7 = (\sqrt{2})^7 \left(\cos\left(-\frac{7\pi}{4}\right) + i\sin\left(-\frac{7\pi}{4}\right)\right)
\]

Notice that \( \cos\left(\frac{7\pi}{4}\right) = \cos\left(-\frac{7\pi}{4}\right) \) and \( \sin\left(\frac{7\pi}{4}\right) = -\sin\left(-\frac{7\pi}{4}\right) \).

Therefore:

\[
(1+i)^7 = (\sqrt{2})^7 \text{cis}\left(\frac{7\pi}{4}\right)
\]
\[
(1-i)^7 = (\sqrt{2})^7 \text{cis}\left(-\frac{7\pi}{4}\right)
\]

Taking the ratio:

\[
\frac{(1+i)^7}{(1-i)^7} = \frac{\text{cis}\left(\frac{7\pi}{4}\right)}{\text{cis}\left(-\frac{7\pi}{4}\right)} = \text{cis}\left(\frac{7\pi}{4} + \frac{7\pi}{4}\right) = \text{cis}\left(\frac{14\pi}{4}\right) = \text{cis}\left(\frac{7\pi}{2}\right) = \text{cis}\left(\frac{3\pi}{2}\right)
\]

This simplifies to:

\[
\frac{(1+i)^7}{(1-i)^7} = -i
\]

### Step 2: Simplify \(\frac{(1-i\sqrt{3})^3}{(1+i\sqrt{3})}\)

Write \(1 - i\sqrt{3}\) and \(1 + i\sqrt{3}\) in polar form:

\[
|1 - i\sqrt{3}| = |1 + i\sqrt{3}| = \sqrt{1^2 + (\sqrt{3})^2} = 2
\]

The arguments are \(\theta = -\frac{\pi}{3}\) and \(\theta = \frac{\pi}{3}\) respectively.

So:

\[
1 - i\sqrt{3} = 2 \text{cis}\left(-\frac{\pi}{3}\right)
\]
\[
1 + i\sqrt{3} = 2 \text{cis}\left(\frac{\pi}{3}\right)
\]

\[
(1 - i\sqrt{3})^3 = 8 \text{cis}\left(-\pi\right) = -8
\]

The final ratio is:

\[
\frac{-8}{2 \text{cis}\left(\frac{\pi}{3}\right)} = -4 \text{cis}\left(-\frac{\pi}{3}\right) = -4\left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right) = -4\left(\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = -2 + 2i\sqrt{3}
\]

### Step 3: Combine the results
Finally, multiply the simplified results:

\[
-i \cdot (-2 + 2i\sqrt{3}) = -i(-2) + (-i)(2i\sqrt{3}) = 2i - 2\sqrt{3}(-1) = 2i + 2\sqrt{3}
\]

So the final simplified form is:

\[
2\sqrt{3} + 2i
\]

Would you like more details or have any questions? Here are 5 related questions to further explore the topic:

1. How do you convert a complex number to polar form?
2. What are the properties of powers of complex numbers?
3. Can you explain the significance of Euler's formula in simplifying complex expressions?
4. How do angles and magnitudes interact in complex number multiplication?
5. What is the geometric interpretation of the multiplication of complex numbers?

**Tip:** Remember that complex numbers can often be simplified by converting them to polar form, especially when dealing with powers and roots.

Learn how to simplify the complex expression (1+i)^7 / (1-i)^7 * (1-i√3)^3 / (1+i√3). Detailed step-by-step solution using De Moivre's theorem and polar form of complex numbers. Suitable for advanced high school students and learners interested in complex number arithmetic.

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