The problem reads:

"Berechnen und vereinfachen Sie so weit wie möglich:"
(Calculate and simplify as far as possible.)

The expression given is:

$\sqrt{a + b} - \sqrt{a} - \sqrt{b}$

Solution:

To simplify $\sqrt{a + b} - \sqrt{a} - \sqrt{b}$, let's analyze the structure.

Unfortunately, there isn't a straightforward algebraic rule to break down the sum inside the square root, i.e., $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. However, we can check specific cases and see if any factors or special identities fit.

Checking special cases or factorizations:

Let’s check if we can rewrite the expression in a useful way. The key here is whether values of $a$ and $b$ satisfy any identities that result in cancellations or simplification:

• If $a = b$, the expression becomes: $\sqrt{2a} - \sqrt{a} - \sqrt{a} = \sqrt{2a} - 2\sqrt{a}.$ This can be rewritten as: $\sqrt{a}(\sqrt{2} - 2).$

• However, for general $a$ and $b$, no further simplification is generally possible beyond the original expression.

Thus, unless specific values are given for $a$ and $b$, the expression $\sqrt{a + b} - \sqrt{a} - \sqrt{b}$ is already as simplified as it can be.

Would you like to explore more specific cases, such as setting particular values for $a$ and $b$? Or do you need further explanations about square roots and simplification methods?

5 Related Questions:

1. What are common simplification rules for square roots?
2. How does one expand and simplify radicals involving sums?
3. What identities are useful in algebra when dealing with square roots?
4. How can you factor or rewrite expressions involving radicals?
5. Are there any geometric interpretations of the expression involving $\sqrt{a + b}$?

Tip: When working with square roots involving sums, it’s often helpful to look for factorizations or substitutions that make the expression simpler.