# Modular Arithmetic Inverse Exponent Simplification

Need help on where to start here.

Given $a^b\mod c = d$, where $b$, $c$ and $d$ is known, how do I find $a$?

Thank!

I just wrote some arbitrary number here:

$x^{13} \mod 47 = 17$, how do I convert this to x = .... mod 47? From my understanding, it will be $x= 7^{1/13} \mod 47$ but how do I simplify that to an all-integer form.

• Do we have any constraints on the values of $b,c,$ and $d$? – Mike Pierce Jan 14 '15 at 16:53
• We have a constraints on the number of digit available on the calculator used to perform the computation to find a. @mapierce271 – Stupid Jan 14 '15 at 17:19

You do the same thing as if this was an equation in more familiar arithmetic: compute the $b$-th roots of $d$ modulo $c$.
• I chose an arbitrary number here $x^{13} \mod 47 = 17$, how do I convert this to x = .... mod 47? Because in this case it will be $x= 7^{1/13} \mod 47$ how do I simplify that? @Hurkyl – Stupid Jan 14 '15 at 22:29
• What confuse me here is the exponent $1/13$, what can we do with it to simplify the expression? Cause if there is a rule for it, please do tell. – Stupid Jan 14 '15 at 22:36