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**Arătați că** \( (1 - i)^4 = -4 \), which translates to **"Show that \( (1 - i)^4 = -4"**.

To prove this, let's work through the complex number \( (1 - i)^4 \).

### Step-by-Step Solution:

1. **Express the number in polar form**:  
   The complex number \( z = 1 - i \).  
   We can write any complex number in the form \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument.

2. **Find the modulus \( r \)**:
   \[
   r = |1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
   \]

3. **Find the argument \( \theta \)**:
   The argument \( \theta \) is the angle the complex number makes with the positive real axis. In this case:
   \[
   \tan \theta = \frac{-1}{1} = -1
   \]
   So, \( \theta = -\frac{\pi}{4} \) (since the complex number is in the fourth quadrant).

   Therefore, in polar form, we have:
   \[
   1 - i = \sqrt{2} \left( \cos \left( -\frac{\pi}{4} \right) + i \sin \left( -\frac{\pi}{4} \right) \right)
   \]

4. **Apply De Moivre's Theorem**:  
   To raise the complex number to the power of 4, apply De Moivre's Theorem, which states:
   \[
   (r (\cos \theta + i \sin \theta))^n = r^n \left( \cos (n \theta) + i \sin (n \theta) \right)
   \]
   In this case, \( n = 4 \), so:
   \[
   (1 - i)^4 = (\sqrt{2})^4 \left( \cos \left( 4 \cdot \left( -\frac{\pi}{4} \right) \right) + i \sin \left( 4 \cdot \left( -\frac{\pi}{4} \right) \right) \right)
   \]

5. **Simplify the expressions**:
   \[
   (\sqrt{2})^4 = 4
   \]
   \[
   4 \cdot \left( -\frac{\pi}{4} \right) = -\pi
   \]
   So:
   \[
   (1 - i)^4 = 4 \left( \cos(-\pi) + i \sin(-\pi) \right)
   \]

6. **Evaluate the trigonometric functions**:
   \[
   \cos(-\pi) = -1, \quad \sin(-\pi) = 0
   \]
   Therefore:
   \[
   (1 - i)^4 = 4 \cdot (-1 + 0i) = -4
   \]

Thus, we have shown that \( (1 - i)^4 = -4 \).

Would you like further details or have any questions?

### Related Questions:
1. How do you find the modulus of any complex number?
2. What is De Moivre's theorem, and how does it apply to powers of complex numbers?
3. How do you convert a complex number into polar form?
4. What is the geometric interpretation of multiplying complex numbers?
5. How can the argument of a complex number be determined from its rectangular coordinates?

**Tip:** De Moivre's theorem is very useful when dealing with powers and roots of complex numbers, especially when using their polar form!

Explore the step-by-step proof of the equation (1 - i)^4 = -4 using complex numbers, polar form, and De Moivre's Theorem. This detailed explanation covers how to express a complex number in polar form and apply the theorem to compute the fourth power.

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