https://file.aiquickdraw.com/mathbot/file/1724102154697t1o5d7ju.jpg

The expression in the image is:

\[
\frac{5\sqrt{2}}{9\sqrt{11} + 2\sqrt{13}}
\]

To simplify this, we should rationalize the denominator. Here are the steps:

### Step 1: Multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of \(9\sqrt{11} + 2\sqrt{13}\) is \(9\sqrt{11} - 2\sqrt{13}\). Multiply both the numerator and denominator by this conjugate:

\[
\frac{5\sqrt{2}}{9\sqrt{11} + 2\sqrt{13}} \times \frac{9\sqrt{11} - 2\sqrt{13}}{9\sqrt{11} - 2\sqrt{13}}
\]

This results in:

\[
\frac{5\sqrt{2} \cdot (9\sqrt{11} - 2\sqrt{13})}{(9\sqrt{11} + 2\sqrt{13})(9\sqrt{11} - 2\sqrt{13})}
\]

### Step 2: Simplify the numerator.
Distribute \(5\sqrt{2}\) across the terms in the numerator:

\[
5\sqrt{2} \cdot 9\sqrt{11} - 5\sqrt{2} \cdot 2\sqrt{13} = 45\sqrt{22} - 10\sqrt{26}
\]

So, the numerator becomes \(45\sqrt{22} - 10\sqrt{26}\).

### Step 3: Simplify the denominator.
The denominator is a difference of squares:

\[
(9\sqrt{11})^2 - (2\sqrt{13})^2 = 81 \cdot 11 - 4 \cdot 13 = 891 - 52 = 839
\]

### Final Expression:

The simplified expression is:

\[
\frac{45\sqrt{22} - 10\sqrt{26}}{839}
\]

Would you like more details on this solution or have any questions?

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Here are 5 related questions:
1. How do you rationalize the denominator for complex fractions?
2. What is the significance of the conjugate when simplifying expressions?
3. Can this technique be applied to non-radical denominators?
4. What happens if the denominator is already rational?
5. How would you simplify an expression involving cube roots?

**Tip:** When rationalizing denominators, always consider using the conjugate for expressions with multiple terms to eliminate radicals effectively.

Learn how to simplify a complex fraction by rationalizing the denominator. This detailed solution covers the steps to rationalize denominators involving radicals, using the conjugate to eliminate radicals effectively. Suitable for high school students and those preparing for standardized tests.

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