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$$  $$löse die wurzel auf indem du ^1/n anwendest

To solve the given expression inside the radical, we'll follow these steps:

1. **Expression under the root**: 
   \[
   \sqrt[z]{a^{-n} \cdot c \cdot a}
   \]
   
2. **Simplify the expression inside the radical**:
   Combine the like terms $a^{-n} \cdot a$:
   \[
   a^{-n+1} \cdot c
   \]

3. **Apply the $z$th root**:
   To remove the radical, raise the expression to the power $ \frac{1}{z} $:
   \[
   (a^{-n+1} \cdot c)^{\frac{1}{z}}
   \]

4. **Distribute the exponent**:
   Apply the exponent $ \frac{1}{z} $ to each factor inside the parentheses:
   \[
   a^{\frac{-n+1}{z}} \cdot c^{\frac{1}{z}}
   \]

So, the simplified expression after applying the $ \frac{1}{z} $ power is:
\[
a^{\frac{-n+1}{z}} \cdot c^{\frac{1}{z}}
\]

Would you like more details or have any questions? 

Here are some related questions to explore:

1. What happens if $n = 1$ in the original expression?
2. How would the expression change if $c$ was raised to a power as well?
3. How would you interpret this expression if $z = 2$?
4. What if $a = 1$? How does that simplify the expression?
5. How would this process differ if $a^{-n} \cdot c \cdot a$ was inside a cube root?

**Tip**: Always combine like terms inside the radical before applying the root to simplify the expression effectively.

Learn how to solve a radical expression by applying the nth root. This problem covers exponentiation, radicals, and fractional powers. Suitable for advanced high school students and anyone interested in understanding radical expressions.

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