Vibrant Colours of Powers

${ 18 }^{ 343/7 }\cdot { 3 }^{ 12345/15 }\\ ={ 18 }^{ 49 }\cdot { 3 }^{ 823 }\\ ={ 3 }^{ 49 }\cdot { 3 }^{ 49 }\cdot { 2 }^{ 49 }\cdot { 3 }^{ 823 }\\ ={ 3 }^{ 921 }\cdot { 2 }^{ 49 }$

Now one thing to note is that every digit from 0 to 9 has a pattern of 4 digits when raised to an exponential power. So for 3, it is:

For ${ 3 }^{ 1 }$ last digit = $3$

For ${ 3 }^{ 2 }$ last digit = $9$

For ${ 3 }^{ 3 }$ last digit = $7$

For ${ 3 }^{ 4 }$ last digit = $1$

And for 2 it is:

For ${ 2 }^{ 1 }$ last digit = $2$

For ${ 2 }^{ 2 }$ last digit = $4$

For ${ 2 }^{ 3 }$ last digit = $8$

For ${ 2 }^{ 4 }$ last digit = $6$

So last digits of ${ 3 }^{ 921 }$ and ${ 2 }^{ 49 }$ (found by using mod) are 3 and 2. So since, $3\cdot 2$ is $6$ , the last digit is $6$