Question

simplify 213 in base 4 multiply by 23 in base 4?

Answer 1

213 in base 4 ---> 2(4^2) + 1(4) + 3 = 39 base 10
23 in base 4 ----> 2(4) + 3 = 11
then 11*39 = 429

429 = 6(64) + 2(16) + 3(4) + 1
= 6(4^3) + 2(4^2) + 3(4) + 1
= 6231 in base 4

Unless you create yourself an addition table
and a multiplication table in base 4, the above method
is probably your best bet.
If you have the tables, you can do the calculation using
our standard algorithm for multiplication

Answer 2

oops, of course there is no 6 in base 4, my bad
so 429 in base 4
= 1(256) + 2(64) + 2(16) + 3(4) + 1
= 12231 in base 4