To find the fifth root of a complex number $Z = -5 + 7i$, we can follow these steps:

Step 1: Express $Z$ in Polar Form

First, we need to convert the complex number $Z = -5 + 7i$ to its polar form. The polar form of a complex number is given by:

$Z = r(\cos \theta + i \sin \theta)$

where:

• $r = |Z|$ is the modulus of the complex number, and
• $\theta = \arg(Z)$ is the argument (angle) of the complex number.

Modulus $r$:

$r = \sqrt{(-5)^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74}$

Argument $\theta$:

$\theta = \tan^{-1}\left(\frac{7}{-5}\right)$ Since the complex number lies in the second quadrant (because $-5$ is negative and $7$ is positive), we adjust the angle accordingly:

$\theta = \pi + \tan^{-1}\left(\frac{7}{-5}\right)$

Step 2: Calculate the Fifth Root

The fifth root of a complex number in polar form is given by:

$Z^{\frac{1}{5}} = r^{\frac{1}{5}} \left[\cos\left(\frac{\theta + 2k\pi}{5}\right) + i\sin\left(\frac{\theta + 2k\pi}{5}\right)\right]$

for $k = 0, 1, 2, 3, 4$.

Modulus of the Fifth Root:

$r^{\frac{1}{5}} = (\sqrt{74})^{\frac{1}{5}}$

Argument of the Fifth Root:

For each $k$, we calculate:

$\theta_k = \frac{\theta + 2k\pi}{5}$

Step 3: Compute Each of the Five Roots

To fully determine the fifth root, you would calculate the above for each value of $k$ (from 0 to 4).

Let's proceed to the calculation for the specific values.

Summary Calculation:

For $Z = -5 + 7i$:

1. $r = \sqrt{74}$
2. $\theta = \pi + \tan^{-1}(-\frac{7}{5})$ radians.

Then, for each $k$ from 0 to 4, compute:

[ Z_k^{1/5} = \sqrt[10]{74} \left[\cos\left(\frac{\theta + 2k\pi}{5}\right) + i \sin\left(\frac{\theta + 2k\pi}{5}\right)\right] ]

This gives you the five distinct roots of $Z$.

Would you like to see a detailed calculation for one specific root or all of them?

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Here are 5 related questions to explore:

1. How do you convert a complex number to its exponential form?
2. What are the properties of roots of complex numbers?
3. How do the roots of a complex number relate geometrically in the complex plane?
4. Can you derive the formula for the roots of a complex number?
5. How does the argument change when a complex number lies in different quadrants?

Tip: Understanding the polar form of complex numbers makes it easier to work with operations like roots and powers.