If a and b are rational numbers and (6 + sqrt(3))/6-s qrt(3) = a + b * sqrt(3) , find the values of a and b.

We are tasked with finding the values of \( a \) and \( b \) if \( \frac{6 + \sqrt{3}}{6 - \sqrt{3}} = a + b\sqrt{3} \), where \( a \) and \( b \) are rational numbers.

### Step 1: Rationalizing the denominator
To simplify the expression on the left-hand side, we will rationalize the denominator. Multiply both the numerator and denominator by the conjugate of the denominator:

\[
\frac{6 + \sqrt{3}}{6 - \sqrt{3}} \cdot \frac{6 + \sqrt{3}}{6 + \sqrt{3}} = \frac{(6 + \sqrt{3})^2}{(6 - \sqrt{3})(6 + \sqrt{3})}
\]

### Step 2: Simplify the denominator
Use the difference of squares formula:

\[
(6 - \sqrt{3})(6 + \sqrt{3}) = 6^2 - (\sqrt{3})^2 = 36 - 3 = 33
\]

So, the denominator is 33.

### Step 3: Expand the numerator
Now, expand \( (6 + \sqrt{3})^2 \):

\[
(6 + \sqrt{3})^2 = 6^2 + 2(6)(\sqrt{3}) + (\sqrt{3})^2 = 36 + 12\sqrt{3} + 3 = 39 + 12\sqrt{3}
\]

### Step 4: Write the fraction
The expression becomes:

\[
\frac{39 + 12\sqrt{3}}{33} = \frac{39}{33} + \frac{12\sqrt{3}}{33}
\]

Simplify each term:

\[
= \frac{39}{33} + \frac{12}{33}\sqrt{3} = \frac{13}{11} + \frac{4}{11}\sqrt{3}
\]

### Step 5: Compare with \( a + b\sqrt{3} \)
Now, compare the result \( \frac{13}{11} + \frac{4}{11}\sqrt{3} \) with \( a + b\sqrt{3} \):

- \( a = \frac{13}{11} \)
- \( b = \frac{4}{11} \)

Thus, the values of \( a \) and \( b \) are:

\[
a = \frac{13}{11}, \quad b = \frac{4}{11}
\]

Would you like any further clarification or details on these steps? 

Here are 5 related questions you might find interesting:

1. How do you rationalize the denominator in expressions involving square roots?
2. Can you simplify the expression \( \frac{2 + \sqrt{5}}{3 - \sqrt{5}} \)?
3. What are conjugates, and how are they used to simplify irrational expressions?
4. How can you expand binomials like \( (a + b)^2 \) using the binomial theorem?
5. Why does rationalizing the denominator help in simplifying expressions?

**Tip:** When working with radicals, multiplying by the conjugate is a useful trick to eliminate the square roots from the denominator.

If a and b are rational numbers and (6 + sqrt(3))/(6 - sqrt(3)) = a + b * sqrt(3), find the values of a and b.

Explore how to find the values of a and b in the equation (6 + sqrt(3))/(6 - sqrt(3)) = a + b * sqrt(3). This problem involves algebraic concepts such as rationalization of radicals using conjugates and binomial expansion. Suitable for students in Grades 10-12.

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