Question

If $2^x-2^{x-2}=192$, then the value of $x$ is

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is D $8$

The given expression $2^x-2^{x-2}=192$ can be simplified as follows:

$2^x-2^{x-2}=192\Rightarrow 2^x(1-2^{-2})=192\Rightarrow 2^x(1-\frac{1}{2^2})=192(\because \frac{1}{x}=x^{-1})\Rightarrow 2^x(1-\frac{1}{4})=192$

$\Rightarrow 2^x\left(\frac{4-1}{4}\right)=192\Rightarrow 2^x\left(\frac{3}{2^2}\right)=192\Rightarrow 3(2^x\times 2^{-2})=3\times 2^6(\because \frac{1}{x}=x^{-1})\Rightarrow 3\times 2^{x-2}=3\times 2^6(\because x^m\times x^n=x^{m+n})$

$\Rightarrow 2^{x-2}=2^6\Rightarrow x-2=6\Rightarrow x=6+2\Rightarrow x=8$

Hence, $x=8$