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The problem provides two complex numbers, \( z_1 \) and \( z_2 \), with known arguments:

1. \( \text{Arg}(z_1) = \frac{\pi}{4} \)
2. \( \text{Arg}(z_2) = 3 \)

We need to find the argument of \( -2 \frac{z_1^4}{z_2^{10}} \).

### Solution Steps

1. **Argument of \( z_1^4 \):**
   Using the property \( \text{Arg}(z^n) = n \cdot \text{Arg}(z) \), we get:
   \[
   \text{Arg}(z_1^4) = 4 \cdot \text{Arg}(z_1) = 4 \cdot \frac{\pi}{4} = \pi.
   \]

2. **Argument of \( z_2^{10} \):**
   Similarly,
   \[
   \text{Arg}(z_2^{10}) = 10 \cdot \text{Arg}(z_2) = 10 \cdot 3 = 30.
   \]

3. **Argument of \( \frac{z_1^4}{z_2^{10}} \):**
   Using the property \( \text{Arg}\left(\frac{z_1}{z_2}\right) = \text{Arg}(z_1) - \text{Arg}(z_2) \), we have:
   \[
   \text{Arg}\left(\frac{z_1^4}{z_2^{10}}\right) = \text{Arg}(z_1^4) - \text{Arg}(z_2^{10}) = \pi - 30 = \pi - 30.
   \]

4. **Argument of \( -2 \):**
   Since \( -2 \) is a real number and lies on the negative real axis, its argument is \( \pi \).

5. **Combining all arguments:**
   Finally, the argument of \( -2 \frac{z_1^4}{z_2^{10}} \) is:
   \[
   \text{Arg}\left(-2 \frac{z_1^4}{z_2^{10}}\right) = \text{Arg}(-2) + \text{Arg}\left(\frac{z_1^4}{z_2^{10}}\right) = \pi + (\pi - 30) = 2\pi - 30.
   \]

Thus, the answer is:
\[
\text{Arg}\left(-2 \frac{z_1^4}{z_2^{10}}\right) = 2\pi - 30.
\]

Would you like a deeper explanation or have any questions?

### Related Questions
1. What is the argument of a product of two complex numbers?
2. How does multiplying a complex number by a negative real number affect its argument?
3. How do we find the argument of a complex number raised to a power?
4. How does division affect the argument of two complex numbers?
5. What is the significance of the principal argument in complex numbers?

### Tip
Always remember that arguments in complex numbers are often considered modulo \( 2\pi \) to keep them within the principal range.

Givet to komplekse tal z1 og z2. Det oplyses at Arg(z1) = π/4 og Arg(z2) = 3. Bestem Arg(-2z1^4/z2^10).

This problem involves calculating the argument of the expression -2z1^4 / z2^10 given that Arg(z1) = π/4 and Arg(z2) = 3. Learn to apply properties of complex numbers, including argument calculations for powers and quotients, suitable for advanced high school math students.

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