$6.(17$ 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 $=10111110111000000000000000000000$

Sign $=1$

Exponent $=125127=-2$

Mantissa $=1\mathrm{.}10000000*2^-2$

Adjusting B by $4=-0\mathrm{.}00011000*2^2$

Adding mantissas $=1\mathrm{.}01100000-0\mathrm{.}00011000=1\mathrm{.}01011000*2^2$

Sign $=0$

Exponent $=2+127=129$

Mantissa $=010110000000000000000000$

Result $=01000000101010000000000000000000$

a $\backslash$times b

Multiplying Mantissas $=1\mathrm{.}01100000*1\mathrm{.}10000000=11\mathrm{.}1100000$

Normalize $=1\mathrm{.}11100000$

Add exponents $=2+(-2)=0$

The sign is negative since one is positive and other is negative

Normalize result $=1.11110000$

Sign $=1$

Exponent $=0+127=127$

Mantissa $=111100000000000000000000$

$754$ Format $=101111111111100000000000000000000$