(6√3 - 4) / (8 - √3) in the form p√3 - q

We are tasked with simplifying the expression \(\frac{6\sqrt{3} - 4}{8 - \sqrt{3}}\) into the form \(p\sqrt{3} - q\).

### Step-by-Step Solution:

1. **Rationalizing the denominator**: 
   To simplify this expression, we will multiply both the numerator and the denominator by the conjugate of the denominator, which is \(8 + \sqrt{3}\). This will eliminate the square root from the denominator.

   The expression becomes:

   \[
   \frac{(6\sqrt{3} - 4)(8 + \sqrt{3})}{(8 - \sqrt{3})(8 + \sqrt{3})}
   \]

2. **Simplifying the denominator**: 
   The denominator is now a difference of squares, which simplifies as:

   \[
   (8 - \sqrt{3})(8 + \sqrt{3}) = 8^2 - (\sqrt{3})^2 = 64 - 3 = 61
   \]

   So the denominator simplifies to 61.

3. **Expanding the numerator**:
   Now we expand the numerator:

   \[
   (6\sqrt{3} - 4)(8 + \sqrt{3}) = 6\sqrt{3}(8 + \sqrt{3}) - 4(8 + \sqrt{3})
   \]
   Expanding both terms:

   \[
   = 6\sqrt{3} \cdot 8 + 6\sqrt{3} \cdot \sqrt{3} - 4 \cdot 8 - 4 \cdot \sqrt{3}
   \]
   \[
   = 48\sqrt{3} + 6 \cdot 3 - 32 - 4\sqrt{3}
   \]
   \[
   = 48\sqrt{3} + 18 - 32 - 4\sqrt{3}
   \]
   Simplify further:
   \[
   = (48\sqrt{3} - 4\sqrt{3}) + (18 - 32)
   \]
   \[
   = 44\sqrt{3} - 14
   \]

4. **Putting it all together**:
   The expression now becomes:

   \[
   \frac{44\sqrt{3} - 14}{61}
   \]

   We can rewrite this as:

   \[
   \frac{44}{61}\sqrt{3} - \frac{14}{61}
   \]

So, the simplified form is:

\[
p\sqrt{3} - q = \frac{44}{61}\sqrt{3} - \frac{14}{61}
\]

### Final Answer:
\[
p = \frac{44}{61}, \quad q = \frac{14}{61}
\]

Would you like more details on any step or have any questions?

Here are 5 related questions:
1. How do you rationalize a denominator with square roots?
2. Why does multiplying by the conjugate remove the square root?
3. Can this technique be used for higher-degree roots?
4. What is the significance of the difference of squares formula?
5. How does the distributive property work with surds?

**Tip**: When dealing with roots in denominators, always look for opportunities to use the conjugate for rationalization.

Learn how to simplify the expression (6√3 - 4) / (8 - √3) into the form p√3 - q using rationalization and the difference of squares. This step-by-step solution is ideal for students in Grades 10-12, covering algebraic concepts and formulas such as surds and the conjugate method.

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