How does an exponent work when it's less than one? I'm rather familiar with exponents, I know that $y^x = y_1 \cdot y_2 \cdot y_3 .... y_x$, but what if the exponent is less than one, how would that work? 
I put in my computer $25^{1/2}$ anyway, expecting it to give me an error, and I got an answer!! And even more surprising, when I did this with more numbers and a little research, I found that $$x^{1/y} = \sqrt[y] x$$ Is it just me, or can exponents take on the role of square roots?

This isn't really a proof but just another way to look at it.
If you raise $a^b$ to the $n$th power, i.e. $(a^b)^n$, you can calculate this by multiplying the exponents:
$$(a^b)^n = a^{bn}.$$
Now consider the case of $n = 1/b$:
$$(a^b)^{1/b} = a^{b\cdot \frac1b} = a^1 = a.$$
So if I square a number ($b=2$), and then get the number back after I raised it to the $1/2$ power, it sure looks like I took the square root.
In more general terms, raising to the $1/b$th power is the inverse operation of raising to the $b$th power (as long as we stay away from $b=0$).
