Exponential Recap I

$\dfrac{\left(1000^{3x}\cdot 10^{\frac{1}{2}}\cdot 10^{\frac{3}{7}+x}\right)}{100^{x+3}\cdot 100^{\frac{1}{2}+x}}=\dfrac{100^{x+1}}{10^{-x}}$

First all the bases equal.

$\dfrac{\left(10^{3\left(3x\right)}\cdot 10^{\frac{1}{2}}\cdot 10^{\frac{3}{7}+x}\right)}{10^{2\left(x+3\right)}\cdot 10^{2\left(\frac{1}{2}+x\right)}}=\dfrac{10^{2\left(x+1\right)}}{10^{-x}}$

Once all bases are equal (all are 10) , then solve algebraically, following the rules of exponents, and ignoring the base of 10.

$\left(9x+\dfrac{1}{2}+\dfrac{3}{7}+x\right)-\left(2x+6+1+2x\right)=2x+2-\left(-x\right)$

$6x-\frac{85}{14}=3x+2$

$3x=\frac{113}{14}$

$x\ =\ \dfrac{113}{42}$

Substituting the value of $x$ into the original equation.

$11787686347.935892 ≠ 11787686347.935854$

However, while they are not mathematically equal, they are still similar when rounded to a whole integer.

$11787686348 = 117876863478$