\(x = 2^b - [8^{30} + 16^5]\)
\(= 2^b - [(2^3)^{30} + (2^4)^5]\)
\(=2^b - [(2^{90}) +(2^{20})]\)
\(= 2^b - [(2^{20})(2^{70} + 1)]\)
\(2^{70} + 1\approx2^{70}\)
therefore, \(x\approx2^b - [(2^{20})(2^{70})]\)
\(x\approx2^b - 2^{90}\)
for lowest value of \(|x|\), \(b = 90\)
hence, B is the correct answer.

Alternatively,
\(x = 2^b - (8^{30} + 16^5)\)
\(= 2^b - [(2^3)^{30} + (2^4)^5]\)
\(= 2^b - [(2^{90}) +(2^{20})]\)
\(= 2^b - 2^{90} - 2^{20}\)
to find the lowest value of \(|x|\), ignore the smaller exponent.
therefore, \(b = 90\)
hence, B is the correct answer.

\(x = 2^b – (8^{30} + 16^5)\)

\(x = 2^b – (2^{3(30)} + 2^{4(5)})\)

\(x = 2^b – (2^{90} + 2^{20})\)

\(x = 2^b – (2^{20}(2^{70} + 1))\)

\(2^{20}(2^{70} + 1)\) can be written as approximately \(= 2^{20}(2^{70}) = 2^{90}\)

\(x = 2^b - 2^{90}\)

Lowest value of \(|x|\) is \(0\). Hence \(b\) should be equal to \(90\).

\(x = 2^{90} - 2^{90}\)

\(2^b = 2^{90} . b = 90\). Answer (B)...