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The problem visible in the image asks you to simplify the expression by rationalizing the denominator of:

\[
\frac{\sqrt{45} + 5\sqrt{2}}{\sqrt{48} - \sqrt{18}}
\]

### Steps to solve:

1. **Simplify the square roots:**
   - \(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\)
   - \(\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}\)
   - \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\)

2. **Rewrite the expression:**
   \[
   \frac{3\sqrt{5} + 5\sqrt{2}}{4\sqrt{3} - 3\sqrt{2}}
   \]

3. **Rationalize the denominator:** Multiply both the numerator and denominator by the conjugate of the denominator, which is \(4\sqrt{3} + 3\sqrt{2}\).

   \[
   \frac{(3\sqrt{5} + 5\sqrt{2})(4\sqrt{3} + 3\sqrt{2})}{(4\sqrt{3} - 3\sqrt{2})(4\sqrt{3} + 3\sqrt{2})}
   \]

4. **Use the difference of squares formula on the denominator:**
   \[
   (4\sqrt{3})^2 - (3\sqrt{2})^2 = 16 \times 3 - 9 \times 2 = 48 - 18 = 30
   \]
   
5. **Expand the numerator using the distributive property:**
   \[
   (3\sqrt{5})(4\sqrt{3}) + (3\sqrt{5})(3\sqrt{2}) + (5\sqrt{2})(4\sqrt{3}) + (5\sqrt{2})(3\sqrt{2})
   \]
   This gives:
   \[
   12\sqrt{15} + 9\sqrt{10} + 20\sqrt{6} + 15 \times 2 = 12\sqrt{15} + 9\sqrt{10} + 20\sqrt{6} + 30
   \]

6. **Final expression:**
   \[
   \frac{12\sqrt{15} + 9\sqrt{10} + 20\sqrt{6} + 30}{30}
   \]
   You can now simplify the terms if necessary by dividing each term by 30.

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Would you like a further breakdown of any step or the simplified final expression?

Here are five related questions for you to explore:

1. How do you rationalize denominators involving two terms with square roots?
2. Can you explain the difference of squares method?
3. What is the purpose of multiplying by the conjugate in rationalization?
4. How would you simplify a more complex square root expression?
5. How does rationalizing help in simplifying irrational denominators?

**Tip:** When simplifying square roots, always check if the number can be factored into a perfect square times another number.

Simplify by rationalizing the denominator of (sqrt(45) + 5sqrt(2)) / (sqrt(48) - sqrt(18)).

This step-by-step guide helps you simplify the expression (sqrt(45) + 5sqrt(2)) / (sqrt(48) - sqrt(18)) by rationalizing the denominator. Learn key concepts like the difference of squares, square root simplification, and rationalization using conjugates. Suitable for students in Grades 10-12.

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