We are given $z = -2 + j$, and we need to express it in polar form and then find $z^6$, converting the result back into Cartesian form.

Step 1: Express $z$ in polar form

The polar form of a complex number $z = x + jy$ is given by:

$z = r \left( \cos \theta + j \sin \theta \right)$

where:

• $r$ is the modulus of $z$,
• $\theta$ is the argument (angle) of $z$.

For $z = -2 + j$:

• The real part $x = -2$,
• The imaginary part $y = 1$.

1.1 Modulus of $z$ (r)

The modulus $r$ is calculated as:

$r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$

So, $r = \sqrt{5}$.

1.2 Argument of $z$ (θ)

The argument $\theta$ is calculated using:

$\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{1}{-2}\right)$

Since the point $(-2, 1)$ lies in the second quadrant, we adjust the argument:

$\theta = \pi - \tan^{-1}\left(\frac{1}{2}\right)$

Using the approximation $\tan^{-1}(0.5) \approx 0.464$ radians:

$\theta \approx \pi - 0.464 = 2.677 \text{ radians}$

Thus, the polar form of $z$ is:

$z = \sqrt{5} \left( \cos(2.677) + j \sin(2.677) \right)$

Step 2: Find $z^6$ in polar form

In polar form, raising a complex number to a power is straightforward. If $z = r \left( \cos \theta + j \sin \theta \right)$, then:

$z^n = r^n \left( \cos(n\theta) + j \sin(n\theta) \right)$

For $z^6$:

• $r^6 = (\sqrt{5})^6 = 5^3 = 125$,
• $6\theta = 6 \times 2.677 = 16.062 \text{ radians}$.

We subtract multiples of $2\pi$ to bring $6\theta$ within the range $[0, 2\pi]$:

$16.062 - 2\pi \times 2 \approx 16.062 - 12.566 = 3.496 \text{ radians}.$

Thus, the polar form of $z^6$ is:

$z^6 = 125 \left( \cos(3.496) + j \sin(3.496) \right)$

Step 3: Convert $z^6$ back to Cartesian form

We use the approximations $\cos(3.496) \approx -0.939$ and $\sin(3.496) \approx -0.343$:

$z^6 = 125 \left( -0.939 + j(-0.343) \right)$

Multiplying through by 125:

$z^6 = -117.375 - 42.875j$

Thus, the Cartesian form of $z^6$ is:

$z^6 = -117.375 - 42.875j$

Final Answer:

$z^6 = -117.375 - 42.875j$

Do you need more details on any step or have any questions?
Here are 5 related questions:

1. How do you convert a complex number from polar to Cartesian form?
2. What is De Moivre's Theorem, and how is it used for powers of complex numbers?
3. What is the geometric interpretation of the modulus and argument of a complex number?
4. How can you find the argument of a complex number in different quadrants?
5. How does multiplying complex numbers in polar form affect their modulus and argument?

Tip: Always ensure to adjust the argument based on the quadrant where the complex number lies.