$1.(12$ points$)$ Assume that registers $s$0$ and $s$1$ hold the values $0$x$80000000$ and $0$xE$0000000,$ respectively when answering the following four questions.

What is the value of $t$0$ after executing add $t$0,$ $s$0,$ $s$1?$

$s$0=0$x$80000000=10000000000000000000000000000000$

$s$1=0$xE$0000000=11100000000000000000000000000000$

$+-------------------------------------------------------------$

$t$0=0101100000000000000000000000000000=0$x$58000000$

Is the result in $t$0$ the desired result, or has there been overflow after executing the add instruction?

There is an overflow, the result should be a negative

What is the value of $t$0$ after executing sub $t$0,$ $s$0,$ $s$1?$

$s$0=10000000000000000000000000000000$

$-$

$s$1=11100000000000000000000000000000$

$--------------------------------------------------------------$

$t$0=-01100000000000000000000000000000=0$xA$0000000$

Is the result in $t$0$ the desired result, or has there been overflow after executing the sub instruction?

No overflow has occurred

$2.(5$ points$)$ What is $4$FD$2-07$A$3$ when these values represent unsigned $16-$bit hexadecimal numbers? The result should be written in hexadecimal. Show your work.

$4$FD$2=20434$

$07$A$3=1955$

$204341955=18479$

$=482$F

$3.(5$ points$)$ What is $4365-3412$ when these values represent unsigned $12-$bit octal numbers? The result should be written in octal. Show your work.

$4365=100011110101$

$3421=011100001010$

$100011110101$

$-$

$011100001010$

$-------------------------$

$000111101011=0753$

$4.(6$ points$)$ What decimal number does the bit pattern $0\backslash$times $0$D$100000$ represent if it is a floating$-$point number? Use the IEEE $754$ standard and show your work.

Sign bit $=0$ so positive

$0$x$0$D$100000=00001101000100000000000000000000$

Exponent bits $=00011010=13$

$13127($exponent for IEEE $754)=-114$

Mantissa $=000100000000000000000000=0\mathrm{.}5$

Float point $=(-1)^$sign $*2^($exponent$)*1.($fraction$)$

$=(-1)^0*2^(-114)*1\mathrm{.}5=7\mathrm{.}22223729*10^-35$

$5.(5$ points$)$ What is the binary representation of the decimal number $62\mathrm{.}25$ assuming the IEEE $754$ single precision format.

$62/2=31($R $=0),31/2=15($R $=1),15/2=7($R $=1),7/2=3($R $=1),3/2=31($R $=1),1?2=0($R$=1)=111110$

Fraction part $=0\mathrm{.}25*2=0\mathrm{.}5=$ integer part $=0$

$0\mathrm{.}5*2=1\mathrm{.}0,$ integer part $=1$

Combine both $=111110\mathrm{.}01$

Its positive so sign bit $=0$

It moves $5$ spaces to the left to be normal $(1\mathrm{.}111101),$ so

Exponent $=5$

$5+127=132=10000100$

Mantissa $=11111001=11111001000000000000000$

The binary representation of $62\mathrm{.}25=01000010011111001000000000000000$

$6.(17$ points$)$ Given the following $32-$bit binary sequences representing single precision IEEE

$754$ floating point numbers:

a $=01000000110110000000000000000000$

b $=10111110111000000000000000000000$

Perform the following arithmetic and show the final addition and multiplication results in both normalized binary format and IEEE $754$ single$-$precision format. Show your steps $($Do not convert a and b to decimal base, compute the addition and multiplication, then convert the results back to normalized binary and single precision$).$

a $+$ b

A $=01000000110110000000000000000000$

Sign $=0$

Exponent $=129127=2$

Mantissa $=1\mathrm{.}01100000*2^2$

B