Definition used:

Congruence.

Let m be an integer greater than 1. If xand yare integers, then x is congruent to y modulo m if x — y is divisible by m. It can be represented as \(\displaystyle{x}={y}{\left({b}\ mod\ {m}\right)}\). This relation is called as congruence modulo m.

Calculation:

Consider the values and congruence classes modulo m,

\(\displaystyle{\left[{13}\right]}^{{{6}}}\) and m = 15.

It is evident that [x]=[y] if \(\displaystyle{\left[{x}={y}{\left({b}\ mod\ {m}\right)}\right]}\). Since \(\displaystyle{13}=-{2}{\left({b} \ mod \ {15}\right)}\).

Then,

[13]=[-2]

\(\displaystyle{\left[{13}\right]}^{{{6}}}={\left[-{2}\right]}^{{{6}}}\)

\(\displaystyle={\left[{\left(-{2}\right)}^{{{6}}}\right]}\)

=[64]

Since, by the principal or congruence \(\displaystyle{64}={4}{\left({b}\ mod \ {15}\right)}\). Therefore, \(\displaystyle{\left[{13}\right]}^{{{6}}}={\left[{4}\right]}\) in \(\displaystyle{Z}_{{{15}}}\).

Therefore, the value of the expression is [4].