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: $\text{Arg}\left(-2 \frac{z_1^4}{z_2^{10}}\right) = 2\pi - 30.$

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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.