(i) Insert 4 geometric means between 256 and -8.
(ii) Insert 4 geometric means between 3 and 96.

Explanation

To solve the problem of inserting geometric means, we will break it down into two parts as given in the question.

### Part (i): Insert 4 geometric means between 256 and -8.

1. **Identify the terms**: Let the geometric means be \( g_1, g_2, g_3, g_4 \). The sequence will be \( 256, g_1, g_2, g_3, g_4, -8 \).

2. **Define the first term and the common ratio**:
- The first term \( a = 256 \).
- The last term can be expressed as \( ar^5 = -8 \) (since there are 4 means, there are 6 terms total).

3. **Set up the equation**:
\[ 256r^5 = -8 \]

4. **Solve for \( r^5 \)**:
\[ r^5 = \frac{-8}{256} = -\frac{1}{32} \]

5. **Express \(-\frac{1}{32}\) in terms of powers**:
\[ -\frac{1}{32} = -1 \cdot \left(\frac{1}{2}\right)^5 = (-1)^5 \cdot \left(\frac{1}{2}\right)^5 \]

6. **Find \( r \)**:
\[ r = -\frac{1}{2} \]

7. **Calculate the geometric means**:
- \( g_1 = ar = 256 \cdot \left(-\frac{1}{2}\right) = -128 \)
- \( g_2 = ar^2 = 256 \cdot \left(-\frac{1}{2}\right)^2 = 64 \)
- \( g_3 = ar^3 = 256 \cdot \left(-\frac{1}{2}\right)^3 = -32 \)
- \( g_4 = ar^4 = 256 \cdot \left(-\frac{1}{2}\right)^4 = 16 \)

8. **Final sequence**: The geometric means inserted are \( -128, 64, -32, 16 \). Thus, the complete sequence is:
\[ 256, -128, 64, -32, 16, -8 \]

### Part (ii): Insert 4 geometric means between 3 and 96.

1. **Identify the terms**: Let the geometric means be \( g_1, g_2, g_3, g_4 \). The sequence will be \( 3, g_1, g_2, g_3, g_4, 96 \).

2. **Define the first term and the common ratio**:
- The first term \( a = 3 \).
- The last term can be expressed as \( ar^5 = 96 \).

3. **Set up the equation**:
\[ 3r^5 = 96 \]

4. **Solve for \( r^5 \)**:
\[ r^5 = \frac{96}{3} = 32 \]

5. **Express \( 32 \) in terms of powers**:
\[ 32 = 2^5 \]

6. **Find \( r \)**:
\[ r = 2 \]

7. **Calculate the geometric means**:
- \( g_1 = ar = 3 \cdot 2 = 6 \)
- \( g_2 = ar^2 = 3 \cdot 2^2 = 12 \)
- \( g_3 = ar^3 = 3 \cdot 2^3 = 24 \)
- \( g_4 = ar^4 = 3 \cdot 2^4 = 48 \)

8. **Final sequence**: The geometric means inserted are \( 6, 12, 24, 48 \). Thus, the complete sequence is:
\[ 3, 6, 12, 24, 48, 96 \]

### Summary of Results:
- For part (i): The geometric means are \( -128, 64, -32, 16 \).
- For part (ii): The geometric means are \( 6, 12, 24, 48 \).