I’m trying to know how to calculate step by step the next equation applying the module in each operation:
$2^\left(4 \times \frac{6}{8}\right) \pmod{11}$
I know that if I just solve the whole equation and then I apply mod 11, the result is 8.
$4 \times \frac{6}{8}= 3$
$2^3 = 8$
$8 \pmod {11} = 8$
But if I try to do with modular arithmetic is not working for me:
$8^{-1} \pmod{11} = 7$
$6 \times 7 \pmod{11} = 9$
$4 \times 9 \pmod{11} = 3$
$2 ^ 3 \pmod{11} = 9$
Something similar happened when I tried to solve first what is inside the brackets in an expression like this:
$2^\left(4\times 6 \times 8\right) \pmod{11} $
But I changed the way to solve the equation and it works
$\left(\left(2^4\right) ^6\right)^8 \pmod{11} $
Now with the division I have no idea what to do, like in the first example I showed.
Any idea how to solve it?
Your modular computation with exponents is meaningless since, by Euler's formula, $$ a^r\equiv a^{r\bmod\varphi(n)}\mod n, $$ so you should compute the exponents modulo $\varphi(11)=10 $. Unfortunately, $8$ is not a unit module $10$.