### Instruction: Let $z=a+bi$ be the complex number with $\vert z \vert = 5$ and $b > 0$ such that the distance between $(1+2i)z^3$ and $z^5$ is maximized. Compute $z^4.$.### Response: To maximize the distance between two complex numbers, I want them to be in opposite directions from the origin.
That means their arguments (or angles with the positive real axis) should differ by $\pi$ radians.
So I need to find the argument of $(1+2i)z^3$ and $z^5$ and set their difference equal to $\pi.$
To find the argument of a product of complex numbers, I can use the fact that the argument of the product is the sum of the arguments of the factors.
So the argument of $(1+2i)z^3$ is the argument of $1+2i$ plus three times the argument of $z.$
Similarly, the argument of $z^5$ is five times the argument of $z.$
The argument of $1+2i$ is $\tan^{-1}(2/1) = \tan^{-1}(2).$
So I have $\tan^{-1}(2) + 3 \arg z - 5 \arg z = \pi.$
Simplifying, I get $-2 \arg z = \pi - \tan^{-1}(2).$
Dividing by -2, I get $\arg z = \frac{1}{2} \tan^{-1}(2) - \frac{\pi}{2}.$
Now I have the argument of $z,$ but I also need its modulus, which is given as $\vert z \vert = 5.$
To find $z,$ I can use the polar form $z = r(\cos \theta + i \sin \theta),$ where $r$ is the modulus and $\theta$ is the argument.
Plugging in the values, I get $z = 5(\cos (\frac{1}{2} \tan^{-1}(2) - \frac{\pi}{2}) + i \sin (\frac{1}{2} \tan^{-1}(2) - \frac{\pi}{2})).$
This is the exact value of $z,$ but I need to find $z^4.$
To do that, I can use the fact that $(z^k) = r^k(\cos k\theta + i \sin k\theta),$ where $(z^k)$ is the complex conjugate of $z^k.$
This means that the modulus of $z^k$ is $r^k$ and the argument of $z^k$ is $k\theta.$
So $z^4 = 5^4(\cos (4(\frac{1}{2} \tan^{-1}(2) - \frac{\pi}{2})) + i \sin (4(\frac{1}{2} \tan^{-1}(2) - \frac{\pi}{2}))).$
Simplifying, I get $z^4 = 625(\cos (2 \tan^{-1}(2) - 2\pi) + i \sin (2 \tan^{-1}(2) - 2\pi)).$
Using the identity $\cos (\theta - 2\pi) = \cos \theta$ and $\sin (\theta - 2\pi) = \sin \theta,$ I can further simplify this to $z^4 = 625(\cos (2 \tan^{-1}(2)) + i \sin (2 \tan^{-1}(2))).$
This is the exact value of $z^4.$