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The image contains a set of mathematical exercises focused on exponential functions, where the task is to rewrite the given expressions in the form \( a^n \), where \( a \) is a real number greater than zero and \( n \) is an integer.

Let's go through each part and simplify the expressions:

### a) \( (2^x)^3 \)
- Use the property \( (a^m)^n = a^{m \cdot n} \).
- Solution: \( (2^x)^3 = 2^{3x} \).

### b) \( (a^3)^0 \)
- Any number raised to the power of 0 is 1 (i.e., \( a^0 = 1 \) for \( a \neq 0 \)).
- Solution: \( (a^3)^0 = 1 \).

### c) \( (a^4 \cdot a^{-3}) \)
- Use the property \( a^m \cdot a^n = a^{m+n} \).
- Solution: \( a^4 \cdot a^{-3} = a^{4 + (-3)} = a^1 = a \).

### d) \( \left[ (a^{-2})^{27} \right]^4 \)
- First simplify the inner exponentiation, using \( (a^m)^n = a^{m \cdot n} \), and then apply the outer exponent:
- Solution: \( \left[ (a^{-2})^{27} \right]^4 = (a^{-54})^4 = a^{-216} \).

### e) \( \left[ (a^2)^{27} \right]^2 \)
- Again, use the power of a power rule:
- Solution: \( \left[ (a^2)^{27} \right]^2 = (a^{54})^2 = a^{108} \).

### f) \( (a^2)^{x^2} \)
- Simplify using the same property \( (a^m)^n = a^{m \cdot n} \):
- Solution: \( (a^2)^{x^2} = a^{2x^2} \).

### g) \( (a^{2^3})^4 \)
- First calculate \( 2^3 = 8 \), and then apply the power of a power rule:
- Solution: \( (a^8)^4 = a^{32} \).

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Would you like further clarification or more detailed explanations on any of these steps?

### Related Questions:
1. How do you apply the property of exponents to simplify expressions like \( (a^m)^n \)?
2. What are the general rules for multiplying and dividing powers with the same base?
3. Can you explain why any non-zero number raised to the power of zero is equal to 1?
4. How do negative exponents work and how do they affect the base?
5. What is the difference between simplifying a power of a power and a product of powers?

### Tip:
Always check for basic exponent rules (like zero exponent and negative exponent) when simplifying expressions, as they make the process much easier.

Desenvolva as expressões a seguir, escrevendo-as na forma a^n, sendo a um número real maior que zero e n um número inteiro. Exercícios envolvendo exponenciação.

Learn how to simplify various exponential expressions in the form a^n. This problem set covers core rules for exponentiation, including multiplying and raising powers to powers. Ideal for high school students in grades 9-11.

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