Find the mean proportional between :
(i) `6 + 3sqrt(3)` and `8 - 4sqrt(3)`
(ii) `a - b` and a^(3) - a^(2)b.

Answer

To find the mean proportional between the given pairs, we will use the formula for mean proportion. The mean proportional \( x \) between two numbers \( a \) and \( b \) can be calculated using the formula:

\[ x = \sqrt{a \cdot b} \]

### Part (i): Find the mean proportional between \( 6 + 3\sqrt{3} \) and \( 8 - 4\sqrt{3} \)

1. **Identify the values**: Let \( a = 6 + 3\sqrt{3} \) and \( b = 8 - 4\sqrt{3} \).

2. **Apply the mean proportional formula**:
\[ x = \sqrt{(6 + 3\sqrt{3})(8 - 4\sqrt{3})} \]

3. **Multiply the two expressions**:
\[ (6 + 3\sqrt{3})(8 - 4\sqrt{3}) = 6 \cdot 8 + 6 \cdot (-4\sqrt{3}) + 3\sqrt{3} \cdot 8 + 3\sqrt{3} \cdot (-4\sqrt{3}) \]
\[ = 48 - 24\sqrt{3} + 24\sqrt{3} - 36 \]
\[ = 48 - 36 = 12 \]

4. **Take the square root**:
\[ x = \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]

Thus, the mean proportional between \( 6 + 3\sqrt{3} \) and \( 8 - 4\sqrt{3} \) is \( 2\sqrt{3} \).

### Part (ii): Find the mean proportional between \( a - b \) and \( a^3 - a^2b \)

1. **Identify the values**: Let \( a = a - b \) and \( b = a^3 - a^2b \).

2. **Apply the mean proportional formula**:
\[ x = \sqrt{(a - b)(a^3 - a^2b)} \]

3. **Factor \( a^3 - a^2b \)**:
\[ a^3 - a^2b = a^2(a - b) \]

4. **Substitute back into the mean proportional formula**:
\[ x = \sqrt{(a - b)(a^2(a - b))} = \sqrt{(a - b)^2 a^2} \]

5. **Simplify**:
\[ x = (a - b) \cdot a \]

Thus, the mean proportional between \( a - b \) and \( a^3 - a^2b \) is \( a(a - b) \).

### Summary of Solutions:
- Mean proportional between \( 6 + 3\sqrt{3} \) and \( 8 - 4\sqrt{3} \) is \( 2\sqrt{3} \).
- Mean proportional between \( a - b \) and \( a^3 - a^2b \) is \( a(a - b) \).