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The problem states:

Given that \( a \) and \( b \) are positive odd integers, and \( a^b b^a \) is a perfect square, prove that \( ab \) is a perfect square.

### Solution

1. **Step 1:** Since \( a^b b^a \) is a perfect square, it means \( a^b b^a = k^2 \) for some integer \( k \).
   
2. **Step 2:** Let’s analyze the powers of \( a \) and \( b \) in the expression \( a^b b^a \).
   \[
   a^b b^a = a^b \cdot b^a
   \]
   To be a perfect square, all exponents in the factorization of \( a \) and \( b \) must be even.

3. **Step 3:** Assume the prime factorizations:
   \[
   a = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k} \quad \text{and} \quad b = q_1^{f_1} q_2^{f_2} \dots q_m^{f_m}
   \]
   Then:
   \[
   a^b = p_1^{e_1 b} p_2^{e_2 b} \dots p_k^{e_k b}
   \]
   \[
   b^a = q_1^{f_1 a} q_2^{f_2 a} \dots q_m^{f_m a}
   \]

4. **Step 4:** Since \( a^b b^a \) is a perfect square, each exponent in the prime factorization of \( a \) and \( b \) must be even. Therefore, \( e_i b \) and \( f_j a \) must be even for all \( i \) and \( j \).

5. **Step 5:** Because \( a \) and \( b \) are both odd, \( e_i \) and \( f_j \) must be even to satisfy the evenness of \( e_i b \) and \( f_j a \). Therefore, \( a \) and \( b \) are themselves perfect squares.

6. **Step 6:** Since \( a \) and \( b \) are perfect squares, their product \( ab \) is also a perfect square.

Thus, we have proven that \( ab \) is a perfect square.

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Would you like further details or have questions?

### Additional Questions
1. How does the parity (odd/even) of integers influence the solution?
2. Can this proof be generalized for other values of \( a \) and \( b \) (not just odd integers)?
3. Why does the factorization require even exponents to form a perfect square?
4. How do prime factorizations assist in proving statements about powers and products?
5. What would change if \( a \) and \( b \) were even integers?

**Tip:** When dealing with perfect squares in proofs, factorize each term to ensure that all exponents are even, which confirms the square nature.

Given that a and b are positive odd integers, and a^b * b^a is a perfect square, prove that ab is a perfect square.

Learn how to prove that the product ab is a perfect square when given that a^b * b^a is a perfect square. This problem explores number theory concepts, including exponentiation, prime factorization, and properties of perfect squares. Suitable for Grades 10-12.

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