Assume $x$ and $y$ are two negative numbers. ‘The result of the multiplication of $x$ and $y$ with the same positive power is greater than the multiplication of $x$ and $y$ with the same negative power.’

Which of the following statements is true?

Statement-$1$ : If $x={\left(\frac{2}{3}\right)}^4÷{\left(\frac{2}{3}\right)}^2,$ then value of $x^2+2x+3$is $0$

Statement-$2$ : If $(-{\frac{1}{2})}^4\times {\left(-2\right)}^8={\left(-2\right)}^{4x}$ then $x=1$

Shyam is trying to find the general formula to find the sum, $2^{-1}+2^{-2}+2^{-3}+2^{-4}+...+2^{-n}$.

He checked a few patterns.

$2^{-1}+2^{-2}=\frac{3}{4}$,  $2^{-1}+2^{-2}+2^{-3}=\frac{7}{8}$, $2^{-1}+2^{-2}+2^{-3}+2^{-4}=\frac{15}{16}$ and so on.

Which of the following, gives the sum of $n$ terms?

Varsha is trying to solve the puzzle below in which $x$ is a positive integer.

$x^{5x}=(81x{)}^x$

What is the value of $x$?