calculating $1024\times(1.11111111)_{2}$

Question

Question

calculate $1024\times(1.11111111)_{2}$ in terms of power of $10$

My Confusion/Approach

$1024*(1.11111111)_{2}=2^{10}(2-2^{-8})=2^{11}-2^{2}=2^{2}(2^{9}-1)$

I am not getting how $(1.11111111)_{2}=(2-2^{-8})$

I know its basic but can't figure out. Please help me out !

Answer 1

\begin{align}1.11111111_2&=2^0+2^{-1}+2^{-2}+2^{-3}+2^{-4}+2^{-5}+2^{-6}+2^{-7}+2^{-8}\\&=2^{-8}(1+2+2^2+2^3+2^4+2^5+2^6+2^7+2^8)\\&=2^{-8}\frac{2^9-1}{2-1}\\&=2-2^{-8}.\end{align}

Answer 2

Hint: $$(0.11111111)_2=\left(\frac 12+\frac 1{2^2}+\ldots+\frac 1{2^8}\right)_{10}$$

Answer 3

$(1.11111111)_2=(2−2^{−8})$

$(1.11111111)_2=10_2 -0.00000001_2=2-2^{-8}$

like 999=1000-1 simply