To simplify the expression $(w^3)^8 \cdot (w^5)^5$, we use the following exponentiation rules:

1. Power of a power rule: $(a^m)^n = a^{m \cdot n}$.
2. Product of powers rule: $a^m \cdot a^n = a^{m+n}$.

Step-by-step solution:

1. Apply the power of a power rule to each term: $(w^3)^8 = w^{3 \cdot 8} = w^{24}$ $(w^5)^5 = w^{5 \cdot 5} = w^{25}$

2. Now the expression becomes: $w^{24} \cdot w^{25}$

3. Apply the product of powers rule: $w^{24} \cdot w^{25} = w^{24 + 25} = w^{49}$

Final Answer:

$w^{49}$

Would you like a detailed explanation of the rules used or have any follow-up questions?

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Here are five related questions for further exploration:

1. How do you simplify expressions with negative exponents, such as $(w^{-3})^4$?
2. What happens if you divide powers with the same base, e.g., $\frac{w^{10}}{w^6}$?
3. Can you explain the difference between power of a product and power of a quotient?
4. How do the rules for fractional exponents work, such as $(w^{1/2})^4$?
5. What is the general formula for simplifying $(a^m)^n \cdot (a^p)^q$?

Tip: Always ensure bases are the same before applying exponent rules. This avoids errors in simplification!