If the coefficient of $x ^{15}$ in the expansion of $\left(a x^3+\frac{1}{b x^{\frac{1}{3}}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{\frac{1}{3}}-\frac{1}{b x^3}\right)^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a, b) :$

If the $6^{th}$ term in the expansion of the binomial ${\left[ {\frac{1}{{{x^{\frac{8}{3}}}}}\,\, + \,\,{x^2}\,{{\log }_{10}}\,x} \right]^8}$ is $5600$, then $x$ equals to

The coefficient of the term independent of $x$ in the expansion of $(1 + x + 2{x^3}){\left( {\frac{3}{2}{x^2} – \frac{1}{{3x}}} \right)^9}$ is

The term independent of $x$ in the expansion of ${\left( {{x^2} – \frac{{3\sqrt 3 }}{{{x^3}}}} \right)^{10}}$ is

The term independent of $x$ in the expansion of ${\left( {2x + \frac{1}{{3x}}} \right)^6}$ is