Find Re (𝑧), Im (𝑧), 𝑧̅and |𝑧| if 𝑧 = 5𝑖
−7 + 6.

We are given the complex number:

\[
z = 5i - 7 + 6
\]

First, simplify this expression:

\[
z = (-7 + 6) + 5i = -1 + 5i
\]

Now, let's find the following:

1. **Real part, \( \text{Re}(z) \)**:
   The real part of \( z = -1 + 5i \) is \( -1 \).

2. **Imaginary part, \( \text{Im}(z) \)**:
   The imaginary part of \( z = -1 + 5i \) is \( 5 \).

3. **Conjugate of \( z \), \( \bar{z} \)**:
   The conjugate of a complex number \( z = a + bi \) is \( \bar{z} = a - bi \). Thus, the conjugate of \( z = -1 + 5i \) is:
   \[
   \bar{z} = -1 - 5i
   \]

4. **Magnitude of \( z \), \( |z| \)**:
   The magnitude of a complex number \( z = a + bi \) is given by:
   \[
   |z| = \sqrt{a^2 + b^2}
   \]
   Here, \( a = -1 \) and \( b = 5 \). Therefore:
   \[
   |z| = \sqrt{(-1)^2 + (5)^2} = \sqrt{1 + 25} = \sqrt{26}
   \]

### Final Answer:
- \( \text{Re}(z) = -1 \)
- \( \text{Im}(z) = 5 \)
- \( \bar{z} = -1 - 5i \)
- \( |z| = \sqrt{26} \)

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do you visualize complex numbers on the complex plane?
2. What are the properties of complex conjugates in operations?
3. How do we compute the argument (angle) of a complex number?
4. What is the polar form of a complex number?
5. How do complex numbers relate to trigonometric identities?

**Tip**: The magnitude of a complex number represents its distance from the origin on the complex plane.

Learn how to find the real and imaginary parts, conjugate, and magnitude of a complex number using a step-by-step solution. This problem involves complex numbers and their properties, suitable for high school students.

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