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We are given the following equation:

\[
\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} + \sqrt{6}} = a + b\sqrt{30}
\]

where \( a \) and \( b \) are integers. We are asked to find \( ab \).

### Solution:

To solve this, let's simplify the left-hand side by rationalizing the denominator.

#### Step 1: Rationalize the denominator

We multiply the numerator and denominator by \( \sqrt{5} - \sqrt{6} \) (the conjugate of the denominator):

\[
\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} + \sqrt{6}} \times \frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}} = \frac{(\sqrt{5} - \sqrt{6})^2}{(\sqrt{5})^2 - (\sqrt{6})^2}
\]

The denominator becomes:

\[
(\sqrt{5})^2 - (\sqrt{6})^2 = 5 - 6 = -1
\]

The numerator is:

\[
(\sqrt{5} - \sqrt{6})^2 = (\sqrt{5})^2 - 2\sqrt{5}\sqrt{6} + (\sqrt{6})^2 = 5 - 2\sqrt{30} + 6 = 11 - 2\sqrt{30}
\]

Thus, the expression becomes:

\[
\frac{11 - 2\sqrt{30}}{-1} = -11 + 2\sqrt{30}
\]

#### Step 2: Compare with the form \( a + b\sqrt{30} \)

By comparison, we have:

\[
a = -11 \quad \text{and} \quad b = 2
\]

#### Step 3: Calculate \( ab \)

Now, we can find \( ab \):

\[
ab = (-11)(2) = -22
\]

Thus, the value of \( ab \) is \( \boxed{-22} \).

---

Would you like more details on any step? Here are some related questions you might find useful:

1. What are conjugates, and why do we use them in rationalizing denominators?
2. Can this method be used for other radicals?
3. How do we determine the form \( a + b\sqrt{30} \)?
4. What is the general technique for simplifying square roots in algebraic expressions?
5. Can rationalizing the denominator be applied in higher dimensions, such as complex numbers?

**Tip:** Always check the form of your solution after simplifying to ensure it matches the required expression.

Jika bilangan bulat a dan b memenuhi \( \frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} + \sqrt{6}} = a + b\sqrt{30} \), maka ab = ...

Learn how to rationalize the radical expression \( \frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} + \sqrt{6}} \), simplify the denominator, and find the values of integers a and b, such that the expression equals \( a + b\sqrt{30} \). Explore the step-by-step solution including key concepts in rationalizing denominators and working with radicals. Suitable for students in Grades 10-12.

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