Question

show that a natural number n is a perfect square( that is k^2 for some k in N) if and only if every exponent in its prime decomposition is even. State, but do not prove, the corresponding result for powers higher than 2

Answer 1

write the prime factor decomp of \(k\) as \[k=p_1^{\alpha_1}\times p_2^{\alpha_2}\times ...\times p_n^{\alpha_n}\] then square it

Answer 2

is there a way to prove it? without using an example?

Answer 3

the above is not an example, as \(k\) i completely general
when you square it, by the laws of exponents, all the exponents will be even

Answer 4

so the laws of exponents is just saying that when you square k on the left you have to square the terms on the right? right? also, it works when the exponents are all even. eg 50^2 = 2^2 * 5^4...? how does the proof show that is possible?