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You've presented various tasks involving calculation of powers and simplification of expressions. Let's address each in order:

### Aufgabe 1: Calculate the following power values without a calculator, converting decimals to fractions where necessary.

1. **\( 125^{2/3} \)**
   - The cube root of 125 is 5, because \( 5^3 = 125 \).
   - \( 125^{2/3} = (5^3)^{2/3} = 5^{(3 \times 2/3)} = 5^2 = 25 \).

2. **\( 4^{-2.5} \)**
   - Convert the decimal exponent to a fraction: \( -2.5 = -\frac{5}{2} \).
   - \( 4^{-\frac{5}{2}} = \frac{1}{4^{\frac{5}{2}}} = \frac{1}{(4^{2.5})} = \frac{1}{(2^2)^{2.5}} = \frac{1}{2^5} = \frac{1}{32} \).

3. **\( 10000^{1.25} \)**
   - Convert \( 1.25 \) to a fraction: \( 1.25 = \frac{5}{4} \).
   - \( 10000^{1.25} = (10^4)^{\frac{5}{4}} = 10^{5} = 100000 \).

### Aufgabe 2: Simplify the following expressions.

1. **\( a^4 \cdot a^6 \cdot a^8 \)**
   - \( a^{4+6+8} = a^{18} \).

2. **\( (-a)^4 \cdot a^7 \)**
   - \( a^4 \cdot a^7 = a^{11} \) since the negative sign in \( (-a)^4 \) vanishes because 4 is even.

3. **\( -3^{-4} \cdot (-3^5) \)**
   - \( -3^{-4} = -\frac{1}{81} \), and \( -3^5 = -243 \).
   - \( -\left(\frac{1}{81}\right) \cdot (-243) = 3 \).

4. **\( a \cdot a^{3.5} \cdot a^{-6} \)**
   - \( a^{1 + 3.5 - 6} = a^{-1.5} = \frac{1}{a^{1.5}} = \frac{1}{a^{3/2}} \).

5. **\( ((-a)^3)^6 \)**
   - \( (-a)^{18} \), since \( 3 \times 6 = 18 \).

6. **\( 0.4^8 \cdot 0.4^{3/5} \)**
   - Converting 0.4 to \( \frac{2}{5} \): \( \left(\frac{2}{5}\right)^8 \cdot \left(\frac{2}{5}\right)^{3/5} = \left(\frac{2}{5}\right)^{8.6} \).

7. **\( a^{-6} \cdot (a \cdot b)^{-6} \)**
   - \( a^{-6} \cdot a^{-6} \cdot b^{-6} = a^{-12} \cdot b^{-6} \).

8. **\( (a^7 \cdot a^7) \cdot a^{-7} \)**
   - \( a^{14} \cdot a^{-7} = a^7 \).

### Aufgabe 3: Explain why \( a^{-n} = \frac{1}{a^n} \) is defined for \( a \neq 0 \).
- The expression \( a^{-n} \) implies \( a \) raised to the power of \(-n\), which mathematically represents the reciprocal of \( a \) raised to the positive power \( n \). This definition extends the properties of exponents to negative numbers, ensuring consistency in the arithmetic of powers. For \( a \neq 0 \), this avoids division by zero, which is undefined.

Would you like more details on any of these points or have other questions? Here are some further queries to consider:

1. How does the negative sign affect powers in general, and why does it disappear when raised to an even power?
2. Can you explain more about converting decimal exponents to fractions?
3. What is the general rule for multiplying and dividing powers?
4. How do properties of exponents help in simplifying expressions with variables?
5. Why is division by zero undefined in mathematics?

**Tip**: When dealing with powers and roots, particularly for non-integer exponents, consider rewriting expressions in terms of fractions to simplify calculations.

Calculate and simplify powers and explain the properties of negative exponents.

Solve power calculations and simplify expressions involving fractional and negative exponents. Learn how to apply exponent rules to simplify complex algebraic expressions.

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