Find the additive inverse of the complex number:
(i) `-5+7i`
(ii) `(sqrt(6)+5i)(sqrt(6)+5i)`

Answer

To find the additive inverse of the given complex numbers, we follow the definition that the additive inverse of a complex number \( z \) is another complex number \( z_1 \) such that \( z + z_1 = 0 \). This means \( z_1 = -z \).

### Solution:

#### (i) Find the additive inverse of the complex number \( -5 + 7i \).

1. **Identify the complex number**:
The complex number given is \( z = -5 + 7i \).

2. **Calculate the additive inverse**:
The additive inverse \( z_1 \) is calculated as:
\[ z_1 = -z = -(-5 + 7i) \]
This simplifies to:
\[ z_1 = 5 - 7i \]

Thus, the additive inverse of the complex number \( -5 + 7i \) is \( 5 - 7i \).

#### (ii) Find the additive inverse of the complex number \( (\sqrt{6} + 5i)(\sqrt{6} + 5i) \).

1. **Multiply the complex number**:
We first need to compute the product:
\[ z = (\sqrt{6} + 5i)(\sqrt{6} + 5i) = (\sqrt{6})^2 + 2(\sqrt{6})(5i) + (5i)^2 \]
This simplifies to:
\[ z = 6 + 10\sqrt{6}i + 25i^2 \]
Since \( i^2 = -1 \), we have:
\[ z = 6 + 10\sqrt{6}i - 25 \]
Combining the real parts gives:
\[ z = -19 + 10\sqrt{6}i \]

2. **Calculate the additive inverse**:
The additive inverse \( z_1 \) is calculated as:
\[ z_1 = -z = -(-19 + 10\sqrt{6}i) \]
This simplifies to:
\[ z_1 = 19 - 10\sqrt{6}i \]

Thus, the additive inverse of the complex number \( (\sqrt{6} + 5i)(\sqrt{6} + 5i) \) is \( 19 - 10\sqrt{6}i \).

### Summary of Results:
- The additive inverse of \( -5 + 7i \) is \( 5 - 7i \).
- The additive inverse of \( (\sqrt{6} + 5i)(\sqrt{6} + 5i) \) is \( 19 - 10\sqrt{6}i \).