So I am trying to figure out why perfect squares cannot have odd exponents. Here are some examples:
The number 36's prime factorization is $6*6$ which is $2^2 3^2$. This perfect square only has even exponents sure.
Let's take 81. The prime factorization of 81 is $3^4$
But what's the intuitive reason or even proof as to why perfect squares can only have even exponents in its prime factorization?
Also if I were to add an exponent to a perfect square… lets say I add a 2 to the prime factorization of 81 so now the prime factorization is $3^4 2^1$, is this new number a multiple of 81 still? Why? Is 81 still a divisor of this new number (I think the new number is 162). What's the reason?
Best Answer
Suppose that $a=b^2$, both integers greater than $1$. We have some prime factorization of $b=p_1^{a_1}\cdots p_k^{a_k}$. Now, square: $$a=b^2=(p_1^{a_1}\cdots p_k^{a_k})^2=p_1^{2a_1}\cdots p_k^{2a_k}$$