The expression in the image is:

$\frac{5\sqrt{2} + 3\sqrt{27}}{-\sqrt{2} - 4\sqrt{27}}$

Let’s break it down step by step.

Step 1: Simplify the square roots

• $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$.

So the expression becomes: $\frac{5\sqrt{2} + 3(3\sqrt{3})}{-\sqrt{2} - 4(3\sqrt{3})}$ $= \frac{5\sqrt{2} + 9\sqrt{3}}{-\sqrt{2} - 12\sqrt{3}}.$

Step 2: This is the final simplified expression:

$\frac{5\sqrt{2} + 9\sqrt{3}}{-\sqrt{2} - 12\sqrt{3}}.$

Would you like to proceed with any further operations, such as rationalizing the denominator or evaluating numerically?

Questions:

1. Do you need further simplification of the expression?
2. Are you looking for a numerical approximation of this expression?
3. Would you like help with rationalizing the denominator?
4. Do you want to understand how to simplify complex expressions involving square roots?
5. Do you want to solve similar expressions?

Tip:

When simplifying square roots, always look for perfect square factors to simplify them efficiently.