How do I figure out the value of a number raised to a fractional power? For example, if I had something like $9^{1\over 2}$, how should I determine how many times I should use the base (the number being raised to the power of the exponent) as a factor?  I just need something specific and readable.  Should I use a root symbol or something?  I'm asking this because I'm thinking it might be the opposite of a normal power.
Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $? Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?   I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.  Thank you.
Why is the root directory denoted by a / sign? I have done some research about this on Google, but the results were cloudy. Why is the / sign used to denote the root directory. Are there any solid reasons behind it?