Question

If \(z^4=7-5i\), then \(\text{Im}((\bar{z})^4)\) is equal to

• A. 5
• B. 7
• C. -7
• D. -5
• E. 0

Answer 1

The Correct Option is A

We are given that \( z^4 = 7 - 5i \). First, recall that the conjugate of \( z \), denoted \( \overline{z} \), satisfies: \[ \overline{z^4} = (\overline{z})^4 \] Taking the conjugate of both sides of \( z^4 = 7 - 5i \), we get: \[ \overline{z^4} = \overline{7 - 5i} = 7 + 5i \] Thus, we have: \[ (\overline{z})^4 = 7 + 5i \] Now, the imaginary part of \( (\overline{z})^4 \) is the coefficient of \( i \), which is \( 5 \). Therefore: \[ \text{Im} \left( (\overline{z})^4 \right) = 5 \]

The correct option is (A) : \(5\)

Answer 2

We are given that \(z^4 = 7 - 5i\). We want to find the imaginary part of \((\bar{z})^4\).

We know that \((\bar{z})^4 = \overline{z^4}\).

Therefore, \((\bar{z})^4 = \overline{7 - 5i} = 7 + 5i\).

The imaginary part of \((\bar{z})^4\) is the imaginary part of \(7 + 5i\), which is 5.

Therefore, Im((\(\bar{z}\))^4) is equal to 5.