The given expression is:

$\frac{\sqrt{8} + \sqrt{32} - \sqrt{50}}{\sqrt{2} + \sqrt{18}}$

Let's simplify this step by step.

Step 1: Simplify each square root term.

1. $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$
2. $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$
3. $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$
4. $\sqrt{2}$ remains as it is.
5. $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Now, substitute these simplified forms into the expression.

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

Step 2: Combine like terms.

• The numerator becomes: $(2\sqrt{2} + 4\sqrt{2} - 5\sqrt{2}) = 1\sqrt{2}$

• The denominator becomes: $(\sqrt{2} + 3\sqrt{2}) = 4\sqrt{2}$

Thus, the expression simplifies to:

$\frac{\sqrt{2}}{4\sqrt{2}}$

Step 3: Simplify the fraction.

Since $\sqrt{2}$ is present in both the numerator and denominator, they cancel out:

$\frac{1}{4}$

Final Answer:

The simplified expression is $\frac{1}{4}$.

Would you like further details or have any questions?

Here are some related questions:

1. How do you simplify expressions with square roots?
2. What are the properties of square roots that allow us to simplify them?
3. How do you handle fractions with square roots?
4. Can the expression change if different square root values are given?
5. What are the steps to rationalize denominators with square roots?

Tip: When simplifying square roots, always factor numbers to extract perfect squares for easier manipulation.