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The question you uploaded asks to "Compute and give your final answer in the form of \( a + b \sqrt{c} \), where \( a \), \( b \), and \( c \) are natural numbers," and the expression is:

\[
\frac{8 \sqrt{3}}{\sqrt{11} - 3} - \sqrt{33} - \sqrt{3}
\]

Let's solve this step-by-step.

### Step 1: Rationalize the first term
We have the term:

\[
\frac{8 \sqrt{3}}{\sqrt{11} - 3}
\]

To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{11} + 3 \):

\[
\frac{8 \sqrt{3} (\sqrt{11} + 3)}{(\sqrt{11} - 3)(\sqrt{11} + 3)}
\]

The denominator becomes:

\[
(\sqrt{11} - 3)(\sqrt{11} + 3) = 11 - 9 = 2
\]

So, the expression becomes:

\[
\frac{8 \sqrt{3} (\sqrt{11} + 3)}{2} = 4 \sqrt{3} (\sqrt{11} + 3)
\]

Simplifying this:

\[
4 \sqrt{3} \times \sqrt{11} + 4 \sqrt{3} \times 3 = 4 \sqrt{33} + 12 \sqrt{3}
\]

### Step 2: Add and subtract the remaining terms
The expression now is:

\[
4 \sqrt{33} + 12 \sqrt{3} - \sqrt{33} - \sqrt{3}
\]

Group like terms:

\[
(4 \sqrt{33} - \sqrt{33}) + (12 \sqrt{3} - \sqrt{3})
\]

This simplifies to:

\[
3 \sqrt{33} + 11 \sqrt{3}
\]

### Final Answer:
The final answer is:

\[
0 + 3 \sqrt{33} + 11 \sqrt{3}
\]

Thus, the values are:

- \( a = 0 \)
- \( b = 11 \)
- \( c = 3 \)

Would you like further details on any of the steps?

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Here are 5 related questions:
1. How do you rationalize a denominator involving square roots?
2. What are like terms, and how do you combine them?
3. How is the difference of squares formula applied when rationalizing?
4. What are natural numbers, and why are \( a \), \( b \), and \( c \) restricted to them?
5. How does multiplying by the conjugate simplify expressions with radicals?

**Tip:** When rationalizing the denominator, always look for the conjugate of the denominator to simplify square root terms effectively.

Compute and give your final answer in the form of a + b√c, where a, b, and c are natural numbers. The expression is 8√3 / (√11 - 3) - √33 - √3.

Learn how to simplify radicals and rationalize denominators by solving 8√3 / (√11 - 3) - √33 - √3. This detailed step-by-step solution involves rationalization, combining like terms, and applying properties of radicals. Suitable for students in Grades 10-12.

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