To solve the given problem, we need to find the expression that is equivalent to $(2 \cdot 9)^4$.

Step 1: Rewrite the expression

$(2 \cdot 9)^4$ This expression can be rewritten using the property of exponents: $(a \cdot b)^n = a^n \cdot b^n$. So, $(2 \cdot 9)^4 = 2^4 \cdot 9^4$

Step 2: Simplify the options

Now, let's analyze each option:

1. Option 1: $2 \cdot 9^4$

   • This is incorrect because only 9 is raised to the 4th power, while 2 is not.

2. Option 2: $2^4 \cdot 9$

   • This is incorrect because 9 is not raised to the 4th power.

3. Option 3: $2^4 \cdot 9^4$

   • This matches our simplification from Step 1, so this is correct.

4. Option 4: $2^9 \cdot 4^9$

   • This is incorrect because both 2 and 9 are raised to incorrect powers.

Final Answer:

The expression $2^4 \cdot 9^4$ is equivalent to $(2 \cdot 9)^4$.

Would you like any further details or explanations on this problem?

Here are 5 related questions for further exploration:

1. How do you expand $(a \cdot b)^n$ for different values of $a$ and $b$?
2. How do you simplify expressions with powers like $(x \cdot y)^m$?
3. Can you solve a similar problem involving division, like $(2 \div 3)^4$?
4. What is the relationship between powers and multiplication in compound expressions?
5. How would you approach solving $(3 \cdot 5^2)^3$?

Tip: Remember that when applying exponents to products, you distribute the exponent to each factor inside the parentheses.