find the fifth root of Z, given that Z=-5+7i

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.

Learn how to calculate the fifth root of the complex number -5 + 7i with a step-by-step solution. Understand the polar form conversion and the formula for finding roots of complex numbers. Suitable for advanced high school students and anyone interested in complex number mathematics.

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