Show how the decimal number -183 would be represented as an 8-bit two's complement arithmetic

For this question you should firstly lay out your placeholders for an 8 bit binary number, from 2^0 up to 2^7 (1,2,4,8,16,32,64,128). Then you find how 183 as a positive number would be represented in binary. 128 is smaller than 183 so you put a 1 under 128. 128 + 64 = 192 which is bigger than 183 so put a 0 under 64. Repeat this process for all the remaining numbers and you should end up with 10110111.
Now flip the binary bits so you get 01001000 and add 1 to this number. Remember we are working with binary so 0+0 = 0, 1+0= 1, however 1+1 = 0 with a carry of 1. The final result should be 01001001