The problem is written in German and asks to "Calculate and simplify as much as possible":

$a^{\frac{1}{2}} - \sqrt[3]{a^{\frac{3}{2}}}$

Let's solve and simplify this expression step by step:

1. Simplify $a^{\frac{1}{2}}$:

   • $a^{\frac{1}{2}}$ represents the square root of $a$: $a^{\frac{1}{2}} = \sqrt{a}$

2. Simplify $\sqrt[3]{a^{\frac{3}{2}}}$:

   • The cube root of $a^{\frac{3}{2}}$ can be simplified using the property of exponents: $\sqrt[3]{a^{\frac{3}{2}}} = a^{\frac{3}{2} \cdot \frac{1}{3}} = a^{\frac{1}{2}}$
   • This also simplifies to $\sqrt{a}$.

3. Combine the simplified expressions:

   • Now, substitute back into the expression: $a^{\frac{1}{2}} - a^{\frac{1}{2}} = \sqrt{a} - \sqrt{a}$

4. Simplify the subtraction:

   • Since the terms are identical, they cancel out: $\sqrt{a} - \sqrt{a} = 0$

Final Answer: $0$

Would you like a detailed explanation of any of the steps?

Related Questions:

1. How do you simplify expressions with fractional exponents?
2. What is the general rule for combining exponents when taking roots?
3. How does one simplify cube roots of powers?
4. What is the relationship between exponents and roots?
5. How can you verify that two terms with exponents are equivalent?

Tip:

Remember that $a^{\frac{m}{n}}$ represents the $n$-th root of $a^m$, which can be written as $\sqrt[n]{a^m}$.