Find the conjugate and modulus of complex number `7-24i`.

Answer

To find the conjugate and modulus of the complex number \( z = 7 - 24i \), we will follow these steps:

### Step 1: Find the Conjugate
The conjugate of a complex number \( z = a + bi \) is given by \( \overline{z} = a - bi \).

For our complex number:
- Real part \( a = 7 \)
- Imaginary part \( b = -24 \)

Thus, the conjugate is:
\[ \overline{z} = 7 + 24i \]

### Step 2: Find the Modulus
The modulus of a complex number \( z = a + bi \) is calculated using the formula:
\[ |z| = \sqrt{a^2 + b^2} \]

For our complex number:
- \( a = 7 \)
- \( b = -24 \)

Now, we calculate:
\[ |z| = \sqrt{7^2 + (-24)^2} \]
Calculating \( 7^2 \) and \( (-24)^2 \):
\[ 7^2 = 49 \]
\[ (-24)^2 = 576 \]

Now, add these values:
\[ |z| = \sqrt{49 + 576} = \sqrt{625} \]

Finally, calculate the square root:
\[ |z| = 25 \]

### Final Result
- The conjugate of \( 7 - 24i \) is \( 7 + 24i \).
- The modulus of \( 7 - 24i \) is \( 25 \).

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