Let's break down and solve the expression shown in the image step by step:

Given expression:

$\frac{\sqrt{8} - \sqrt{16}}{4 - \sqrt{2}} - 2^{\frac{1}{2}}$

Step 1: Simplify square roots

• $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$
• $\sqrt{16} = 4$
• $\sqrt{2}$ remains as it is.

Now, the expression becomes:

$\frac{2\sqrt{2} - 4}{4 - \sqrt{2}} - 2^{\frac{1}{2}}$

Step 2: Simplify powers

• $2^{\frac{1}{2}} = \sqrt{2}$

Now the expression is:

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

Step 3: Combine terms (if possible)

We'll leave the subtraction of $\sqrt{2}$ until after simplifying the fraction:

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

To simplify the fraction, multiply the numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of $4 - \sqrt{2}$ is $4 + \sqrt{2}$.

Step 4: Multiply by the conjugate

$\frac{(2\sqrt{2} - 4)(4 + \sqrt{2})}{(4 - \sqrt{2})(4 + \sqrt{2})}$

Now, calculate the denominator using the difference of squares formula:

$(4 - \sqrt{2})(4 + \sqrt{2}) = 16 - 2 = 14$

Next, expand the numerator:

$(2\sqrt{2} - 4)(4 + \sqrt{2}) = 2\sqrt{2} \times 4 + 2\sqrt{2} \times \sqrt{2} - 4 \times 4 - 4 \times \sqrt{2}$ $= 8\sqrt{2} + 2 \times 2 - 16 - 4\sqrt{2} = 8\sqrt{2} + 4 - 16 - 4\sqrt{2}$ $= 4\sqrt{2} - 12$

Thus, the fraction simplifies to:

$\frac{4\sqrt{2} - 12}{14} = \frac{2\sqrt{2} - 6}{7}$

Step 5: Subtract $\sqrt{2}$

Now, subtract $\sqrt{2}$:

$\frac{2\sqrt{2} - 6}{7} - \sqrt{2}$

To subtract these terms, rewrite $\sqrt{2}$ as $\frac{7\sqrt{2}}{7}$ to have a common denominator:

$\frac{2\sqrt{2} - 6}{7} - \frac{7\sqrt{2}}{7} = \frac{(2\sqrt{2} - 7\sqrt{2} - 6)}{7}$

$= \frac{-5\sqrt{2} - 6}{7}$

Final Answer:

The simplified expression is:

$\boxed{\frac{-5\sqrt{2} - 6}{7}}$

Do you want more details or have any questions?

Here are some related questions:

1. How do you simplify square roots of non-perfect squares?
2. What is the difference of squares formula, and why is it useful in rationalizing denominators?
3. Can you explain how to simplify radicals with variables?
4. How would the result change if one of the numbers in the radical was different?
5. What are some common ways to simplify complex fractions?

Tip: When working with square roots, always try to factor numbers under the radical into perfect squares to simplify them easily!