Sir Surds

\Large \begin{aligned} \left(\sqrt [\frac{x^5}{5}] {\sqrt [\frac{1}{5x^5}] {\left(5^\sqrt{5x}\right)^{x^5}}} \right)^{5x} & = 5^{\sqrt [5] {5x^5}} \\ \Large \left( \left( \left( \left(5^{5^{0.5}x^{0.5}} \right)^{x^5} \right)^{5x^5} \right)^{5x^{-5}} \right)^{5x} & = 5^{\left(5x^5\right)^{0.2}} \\ \Large 5^{5^{0.5}x^{0.5} \cdot x^5 \cdot 5x^5 \cdot 5x^{-5} \cdot 5x} & = 5^{5^{0.2}x} \\ \Large 5^{5^{0.5+1+1+1}x^{0.5+5+5-5+1}} & = 5^{5^{0.2}x} \\ \Large 5^{5^{3.5}x^{6.5}} & = 5^{5^{0.2}x} \\ \Large \implies 5^{3.5}x^{6.5} & = 5^{0.2}x \\ \Large x^{5.5} & = 5^{-3.3} \\ \Large \implies x & = 5^{-\frac{3.3}{5.5}} = 5^{-\frac{3}{5}} \end{aligned}

$\Large \implies a + b = 3 + 5 = \boxed{8}$

$\begin{aligned} \LARGE {\left(\sqrt[\dfrac{x^5}{5}]{\sqrt[\dfrac{1}{5x^5}]{{\left(5^{\sqrt{5x}}\right)}^{x^5}}}\right)}^{5x} & = \LARGE 5^{\sqrt[5]{5x^5}}\\ \\ \LARGE {\left({\left({\left({\left(5^{5x}\right)}^{x^5}\right)}^{5x^5}\right)}^{\frac{5}{x^5}}\right)}^{5x} & = \LARGE 5^{\sqrt[5]{5}x}\\ \\ \LARGE 5^{\sqrt{5x} × x^5 × 5x^5 × \tfrac{5}{x^5} × 5x} & = \LARGE 5^{\sqrt[5]{5}x}\\ \\ \LARGE 5^{125 × \sqrt{5} × x^{\frac{13}{2}}} & = \LARGE 5^{\sqrt[5]{5}x}\\ \\ \LARGE 5^{5^3 × 5^{\frac{1}{2}} × x^{\frac{13}{2}}} & = \LARGE 5^{5^{\frac{1}{5}} × x}\\ \\ \text{Equating the powers}:-\\ \Large \dfrac{5^3 × 5^{\frac{1}{2}}}{5^{\frac{1}{5}}} & = \Large \dfrac{x}{x^{\frac{13}{2}}}\\ \\ \Large 5^{\tfrac{33}{10}} & = \Large x^{-\tfrac{11}{2}}\\ \\ \text{Raising root of 11 on both sides}:-\\ \Large 5^{\tfrac{3}{10}} & = \Large x^{-\tfrac{1}{2}}\\ \\ \text{Raising power of -2 on both sides}:-\\ \Large 5^{-\tfrac{3}{5}} & = x\\ \\ \therefore a + b & = 3 + 5\\ & = \boxed{8} \end{aligned}$

$Let\ a=5x,\ \ \ b=\dfrac 5 {x^5},\ \ \ c=5x^5,\ \ \ d=x^5,\ \ \ e=5^{0.5}*x^{0.5}.\\ m=5^{0.2}*x*{1 }\\ \text{Since on both the sides the base is 5, we equate the powers as under.}\\ a*b*c*d*e=5^{3.5}*x^{6.5}=5^{0.2}*x.\\ \implies\ x=5^{ - 0.6}=5^{ - \frac 3 5}.\ \ \ \ a+b=3+5=8$