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.