Percentage calculation challenge (2)

$\large 2016\text{\%}\times2015\text{\%}\times2014\text{\%}\times\dots\times3\text{\%}\times2\text{\%}\times1\text{\%} = \dfrac{2016!}{10^{x}\text{\%}}$

$\large (2016\times2015\times2014\times\dots\times3\times2\times1)\times(\underbrace{10^{-2}\times10^{-2}\times10^{-2}\times\dots\times10^{-2}}_{2016\text{times}}) = \dfrac{2016!}{10^{ \color{#3D99F6}{x-2} }}$

$\large 2016!\times 10^{-4032} = \dfrac{2016!}{10^{x-2}}$

$\dfrac{2016!}{10^{4032}} = \dfrac{2016!}{10^{x-2}}$

$x = 4032+2 = 4034$

Note : $\text{\%} = 10^{-2}$

Relevant wiki: Rules of Exponents - Algebraic

We know,

$x\text{\%}=\frac{x}{100}$

$\begin{aligned} \because \quad \quad 2016\text{\%}\times 2015\text{\%}\times 2014\text{\%}\times \dots \times 3\text{\%}\times 2\text{\%}\times 1\text{\%} & =\frac { 2016 }{ 100 } \times \frac { 2015 }{ 100 } \times \frac { 2014 }{ 100 } \times \dots \times \frac { 3 }{ 100 } \times \frac { 2 }{ 100 } \times \frac { 1 }{ 100 } \\ \Longrightarrow 2016\text{\%}\times 2015\text{\%}\times 2014\text{\%}\times \dots \times 3\text{\%}\times 2\text{\%}\times 1\text{\%} & =\frac { 2016\times 2015\times 2014\times \dots \times 3\times 2\times 1 }{ \underbrace { 100\times 100\times 100\times \dots \times 100\times 100 } } \\& \quad \quad \quad \quad \quad \quad \quad \quad 2016\quad times \\ \Longrightarrow 2016\text{\%}\times 2015\text{\%}\times 2014\text{\%}\times \dots \times 3\text{\%}\times 2\text{\%}\times 1\text{\%} & =\frac { 2016! }{ 100^{ 2016 } } \end{aligned}$

Substitute $2016\times 2015\text{\%}\times 2014\text{\%}\times \dots \times 3\text{\%}\times 2\text{\%}\times 1\text{\%} =\frac { 2016! }{ 100^{ 2016 } }$ into

$2016\text{\%}\times 2015\text{\%} \times2014\text{\%}\times \dots \times 3\text{\%}\times 2\text{\%}\times 1\text{\%} = \frac{2016!}{10^{x}\text{\%}}$ we have:

$\large \frac { 2016! }{ 100^{ 2016 } } =\frac { 2016! }{ 10^{ x }\% }$

Cancel out numerators, we have:

$\begin{aligned} 100^{ 2016 } & ={ 10 }^{ x }{ \text{\%} } \\ (10^{ 2 })^{ 2016 } & =\frac { 10^{ x } }{ 100 } \\ 100\times (10^{ 2 })^{ 2016 } & =10^{ x } \\ 10^{ 2 }\times (10^{ 2 })^{ 2016 } & =10^{ x } \\ 10^{ 2 }\times 10^{ 2\times 2016 } & =10^{ x } \\ 10^{ 2+4032 } & =10^{ x } \\ 10^{ 4034 }\quad & =10^{ x } \\ x & =\boxed{4034} \end{aligned}$ .