What is $x^y$? How to understand it? $x+y=z$

I have a pen. He has a pen. Total is two pen. This is plus.

$x-y=z$

I had two pens. A pen was lost. So, I have a pen. Total remaining is one. This is minus.

$x\cdot y=z$

I have two pens. Three other friends have two pens. $4 \text{(persons)}\cdot 2 \text{(pens/person)}=8 \text{pens}$. This is multiply.                         

$x^y=z$

How to understand this?
I have one pen. I become really angry and I break it into $x$ pieces. Each time I get angry I break each of my pieces into $x$ even tinier pieces. I become angry $y$ times. I end up with $x^y$ pieces.
I have two friend. my friend has two more friend each which doesn't know each other, and they have 4 pen each.