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Let's analyze the given problem:

The equation states:
\[
2^a = 3^b
\]
and asks which expression is equivalent to \( 12^{ab} \).

We start by expressing \( 12^{ab} \) in terms of base 2 and base 3:
\[
12^{ab} = (2^2 \cdot 3)^{ab}
\]
Breaking it down:
\[
12^{ab} = (2^2)^{ab} \cdot 3^{ab} = 2^{2ab} \cdot 3^{ab}
\]
Using the given \( 2^a = 3^b \), we can substitute \( 2^a \) for \( 3^b \):
\[
3^{ab} = (2^a)^b = 2^{ab}
\]
So, we have:
\[
12^{ab} = 2^{2ab} \cdot 2^{ab} = 2^{3ab}
\]

Now we check the options:

- **Option A:** \( 2^{a^2} \) — does not match.
- **Option B:** \( 2^{b^2} \) — does not match.
- **Option C:** \( 2^{(a+b)^2} \) — does not match.
- **Option D:** \( 2^{a^2 + 2ab} \) — does not match.

However, **the correct expression for \( 12^{ab} \)** based on our work is \( 2^{3ab} \), and none of the options match this exactly. Could you double-check the problem statement for any additional clarifications?

If you have further questions or would like more details, feel free to ask!

Here are 5 related questions to consider:
1. How do you express a number with multiple prime factors like 12 in terms of prime bases?
2. How does the substitution of variables help simplify exponents?
3. What are the general laws of exponents that apply here?
4. How would the solution change if different bases were used?
5. Why is it helpful to express everything in terms of powers of 2?

**Tip:** When dealing with exponentials, look for ways to express everything with the same base to simplify calculations.

If 2^a = 3^b, then which of the following is equal to 12^ab?

Learn how to solve for 12^ab using the given equation 2^a = 3^b by breaking it down using exponent laws. This problem involves converting expressions to the same base and simplifying using algebraic methods. Suitable for high school students in grades 9-12.

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