Solve the following equations for x:

Solution:

(i)

$2^{2 x}-2^{x+3}+2^{4}=0$

$\Rightarrow\left(2^{x}\right)^{2}-\left(2^{x} \times 2^{3}\right)+\left(2^{2}\right)^{2}=0$

$\Rightarrow\left(2^{x}\right)^{2}-2 \times 2^{x} \times 2^{2}+\left(2^{2}\right)^{2}=0$

$\Rightarrow\left(2^{x}-2^{2}\right)^{2}=0$

$\Rightarrow 2^{x}-2^{2}=0$

$\Rightarrow 2^{x}=2^{2}$

$\Rightarrow x=2$

(ii)

$3^{2 x+4}+1=2.3^{x+2}$

$\Rightarrow\left(3^{x+2}\right)^{2}-2.3^{x+2}+1=0$

$\Rightarrow\left(3^{x+2}-1\right)^{2}=0$

$\Rightarrow 3^{x+2}-1=0$

$\Rightarrow 3^{x+2}=1=3^{0}$

$\Rightarrow x+2=0$

$\Rightarrow x=-2$