The floating point representation of a real number has the form $x=+-(0.d_1{d}_{2}dots{d}_n{)}_{}*{}^e,$ where $d_{1}0,0d_i-1,-MeM.$

a$)$ Suppose $=2,n=4,M=3.$ How many different numbers can be represented in this system?

b$)$ Find $f(\sqrt[\phantom{2}]{2}),$ the floating point representation of $\sqrt[\phantom{2}]{2}$ in this system. Apply the usual rules for rounding numbers up and down. Then express the final result in decimal form. How large is the roundoff error?

c$)$ In class we saw that $x_{\text{m}\text{a}\text{x}}=7\mathrm{.}5,x_{\text{m}\text{i}\text{n}}=0\mathrm{.}0625($these are the largest and smallest positive numbers that can be represented in this system$).$ Suppose we switch to a system with $n=5.$ Find the new values of $x_{\text{m}\text{a}\text{x}},x_{\text{m}\text{i}\text{n}}.$