Sir Surds

Algebra Level 4

\[\LARGE {\left(\sqrt[\dfrac{x^5}{5}]{\sqrt[\dfrac{1}{5x^5}]{{\left(5^{\sqrt{5x}}\right)}^{x^5}}}\right)}^{5x} = 5^{\sqrt[5]{5x^5}}\]

Find the real value of \(x\) satisfying the real equation above.

The answer is of the form \(5^{-\dfrac{a}{b}}\), where \(a\) and \(b\) are co-prime positive integers, then what is the value of \(a + b\)?

\(\text{Note}\): Here \(x \neq \{-1 , 0 , 1\}\)

This is one part of the set Fun with exponents

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