https://file.aiquickdraw.com/mathbot/file/17267530117969chi4uj5.jpg

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.

Which expression is equivalent to (2 · 9)^4?

Discover how to find the expression equivalent to (2 · 9)^4 by applying the distributive property of exponents. This problem covers key exponentiation rules suitable for students in Grades 7-9.

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