The problem involves evaluating two expressions with complex numbers:

1. $(1 + i\sqrt{3})^{3/7}$
2. $(1 - i)^{1/5}$

Step 1: Express in Polar Form

For complex numbers, it's often convenient to rewrite them in polar form as $re^{i\theta}$, where:

• $r$ is the modulus: $r = \sqrt{a^2 + b^2}$
• $\theta$ is the argument: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

For $(1 + i\sqrt{3})$:

• $r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$
• $\theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$

So, $1 + i\sqrt{3} = 2e^{i\pi/3}$.

Raise $(1 + i\sqrt{3})$ to $\frac{3}{7}$:

$(1 + i\sqrt{3})^{3/7} = \left( 2e^{i\pi/3} \right)^{3/7}$ Using the power property of complex numbers: $r^{3/7} e^{i(3\theta/7)} = 2^{3/7} e^{i(3\pi/21)} = 2^{3/7} e^{i\pi/7}$

For $(1 - i)$:

• $r = \sqrt{1^2 + (-1)^2} = \sqrt{2}$
• $\theta = \tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}$

So, $1 - i = \sqrt{2}e^{-i\pi/4}$.

Raise $(1 - i)$ to $\frac{1}{5}$:

$(1 - i)^{1/5} = \left( \sqrt{2}e^{-i\pi/4} \right)^{1/5}$ Using the power property of complex numbers: $r^{1/5} e^{i(\theta/5)} = (\sqrt{2})^{1/5} e^{-i\pi/20}$

Final Results:

1. $(1 + i\sqrt{3})^{3/7} = 2^{3/7} e^{i\pi/7}$
2. $(1 - i)^{1/5} = (\sqrt{2})^{1/5} e^{-i\pi/20}$

Would you like a numerical approximation or further steps?