$[1]$ Assuming $w=4,$ we can assign a numeric value to each possible hexadecimal digit, assuming either an unsigned or a two's$-$complement interpretation. Fill in the $.$llowing table according to these interpretations by writing out the nonzero powers of $2$ in the summations shown in equations below:

$B2T_w(vec(x))-x_{w-1}{2}^{w-1}+{\displaystyle \underset{i=0}{\overset{w-2}{}}}{x}_i{2}^i$

$B2U_w(vec(x)){\displaystyle \underset{i=0}{\overset{w-1}{}}}{x}_i{2}^i$