Question

OwlKid's kiddy proof.
Show that an integer to an even power is a perfect square.

Answer 1

\( \color{Black}{\Rightarrow START}\)

Let's start out with the identity \(\large (a^b)^c = a^{bc}\).

Wait. Let's leave that hanging above for a while. A prerequisite for all people is to know the definition of even numbers. An even number is an integer in the form \(2k: k \in \mathbb{Z}\). Let's assume that we have \( x^{2k}\), where both \(k\) and \(x\) are integers.

\( \color{Black}{\Rightarrow x^{2k} }\)

Use that identity NOW.

\( \color{Black}{\Rightarrow (x^k)^2}\)

Now, if you sense, we have a perfect square. Remember - \((\cdots)^2\) is a perfect square!

HENCE, PROVED.
$HAKA!

Answer 2

Question says "show" not "prove". Not really sure this is a "proof" :P

Answer 3

Additional note: A perfect square is a number which can be expresses as \((\mathbb{Z})^2\), and in order for \((\cdots)^2\) to be a perfect one, \(\cdots\) must be a square.

Answer 4

Well, it is one of them. Showing is just proving ;)

Answer 5

NO ITS NOT @ParthKohli

Answer 6

A debate here? I don't need it.

Answer 7

K iw as kidding SHOWING MEANS PROVING HAPPY!

Answer 8

@uri, you're so cute :P

Answer 9

Well,i am! :D why does everyone say that anyway?

Answer 10

Not everyone knows how be cute, so when they see someone who is, they feel compelled to acknowledge it :P

Answer 11

:|

Answer 12

K thanks @Hero :)

Answer 13

lol how is that not a proof?
I guess you should have started with the words assume or let then it would "look" more like a proof.

Answer 14

That is why I am kiddy :)

Answer 15

i guess you shoult state the universe but outside of that looks fine to me:)

Answer 16

'Kiddy proofs' is like the own series that I started to state axioms. There is no proof. I agree.
I sometimes pity on the people who ask 'why so?' to questions like these.
There's where I decided to make this kind of stuff.

Answer 17

I hope you understand what I feel :)