Is This a Sufficient Proof for Perfect Squares with Even Exponents?

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SUMMARY

The discussion centers on the proof of whether a number \( n \) can be classified as a perfect square if all exponents \( r \) in its prime factorization are even. The prime factorization is represented as \( n = p_1^{r_1} \cdots p_k^{r_k} \). It is established that for \( n \) to be a perfect square, each exponent \( r_i \) must indeed be even. However, the proof lacks clarity and coherence, particularly in defining the variables involved, such as \( p \) and \( r \).

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n=p1r1...pkrk

In order for p to be a perfect square, r must be even. Therefore

n=p12h1...pk2hk

taking the square root of both sides I'm just left with

n=p1h1...pkhk

Does this work as a proof that n is a perfect square if r is even? It's a homework problem and I'm not sure if this is sufficient or correct at all.
 
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If the p's are distinct primes, it's true the r's must be even for n to be a perfect square, but you haven't proved it. You haven't even stated the problem coherently. What's p?
 

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