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Let's go through each part of the exercise using the laws of exponents and radicals.

### 1) Simplify using the laws of exponents:

a) \( 3^3 \cdot 3^4 \cdot 3 \)
   - Use the property of exponents: \( a^m \cdot a^n = a^{m+n} \)
   - \( 3^3 \cdot 3^4 \cdot 3 = 3^{3+4+1} = 3^8 \)

b) \( 5^7 \cdot 5^3 \)
   - \( 5^7 \cdot 5^3 = 5^{7+3} = 5^{10} \)

c) \( (5^3)^4 \)
   - Use the power rule: \( (a^m)^n = a^{m \cdot n} \)
   - \( (5^3)^4 = 5^{3 \cdot 4} = 5^{12} \)

d) \( (5 \cdot 2 \cdot 3)^4 \)
   - Distribute the exponent: \( (a \cdot b)^n = a^n \cdot b^n \)
   - \( (5 \cdot 2 \cdot 3)^4 = 5^4 \cdot 2^4 \cdot 3^4 \)

e) \( (3^4)^4 \)
   - \( (3^4)^4 = 3^{4 \cdot 4} = 3^{16} \)

f) \( \left( [5^3]^4 \right)^2 \)
   - First simplify the inner part: \( (5^3)^4 = 5^{12} \)
   - Now apply the outer exponent: \( (5^{12})^2 = 5^{12 \cdot 2} = 5^{24} \)

g) \( (8^2)^3 \)
   - \( (8^2)^3 = 8^{2 \cdot 3} = 8^6 \)

h) \( (9^3)^2 \)
   - \( (9^3)^2 = 9^{3 \cdot 2} = 9^6 \)

i) \( 2^5 \cdot 2^4 \cdot 2 \)
   - \( 2^5 \cdot 2^4 \cdot 2 = 2^{5+4+1} = 2^{10} \)

j) \( 27^7 \div 27^6 \)
   - Use the division property: \( a^m \div a^n = a^{m-n} \)
   - \( 27^7 \div 27^6 = 27^{7-6} = 27^1 = 27 \)

k) \( (2^2)^4 \)
   - \( (2^2)^4 = 2^{2 \cdot 4} = 2^8 \)

l) \( (4 \cdot 2 \cdot 3)^4 \)
   - \( (4 \cdot 2 \cdot 3)^4 = 4^4 \cdot 2^4 \cdot 3^4 \)

m) \( (2^5)^4 \)
   - \( (2^5)^4 = 2^{5 \cdot 4} = 2^{20} \)

n) \( \left( (2^3)^4 \right)^0 \)
   - Anything raised to the power of 0 is 1, so \( \left( (2^3)^4 \right)^0 = 1 \)

Would you like a detailed explanation of any of the specific steps? Here are 5 questions to deepen your understanding:

1. What is the rule for multiplying powers with the same base?
2. How does the power of a product rule apply when there are multiple terms in parentheses?
3. What happens when you divide powers with the same base?
4. How does the power of a power rule work in exponential expressions?
5. Why does raising any non-zero number to the power of zero result in 1?

**Tip:** When simplifying expressions with multiple powers, always apply exponent rules carefully in the correct order to avoid mistakes.

Simplifica empleando las leyes de los exponentes.

Este ejercicio trata sobre la simplificación de expresiones con potencias y radicales utilizando las leyes de los exponentes, como el producto de potencias, la regla del exponente cero, y más. Aprende cómo aplicar cada regla con ejemplos detallados.

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