In IEEE $754$ standard, a single precision representation uses $32$ bits, in which $1$ bit is used for the sign and $8$ bits are used for the exponent with bias $127.$

$($a$)10\%$ What is the corresponding decimal number for the following data with IEEE $754$ standard: $11000001011010000000000000000000?$ You have to show your derivation.

$($b$)10\%$ What is the corresponding IEEE $754$ single representation for the decimal number $5\mathrm{.}8125?$ You have to show your derivation.