Nested roots

Algebra Level 3

x 2 x 3 x 4 x 5 x = x 5 x 4 x 3 x 2 x \Large \sqrt[5x]{\sqrt[4x]{\sqrt[3x]{\sqrt[2x]{x}}}} = \sqrt[x]{\sqrt[x^2]{\sqrt[x^3]{\sqrt[x^4]{x^5}}}}

Find the real value of x x satisfying the real equation above.

The answer is of the form a 6 \sqrt[6]{a} , then submot the value of a a .

Note:- x ≠ 1 x \neq 1

This is one part of the set Fun with exponents


The answer is 600.

Ashish Menon by Ashish Menon , 20, India

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1 solution

Ashish Menon
Apr 25, 2016

x 2 x 3 x 4 x 5 x = x 5 x 4 x 3 x 2 x ( ( ( x 1 2 x ) 1 3 x ) 1 4 x ) 1 5 x = ( ( ( x 5 1 x 4 ) 1 x 3 ) 1 x 2 ) 1 x x 1 120 x 4 = x 5 x 10 Equating the powers : − 1 120 x 4 = 5 x 10 x 6 = 120 × 5 x 6 = 600 x = 600 6 ∴ a = 600 \begin{aligned} \sqrt[5x]{\sqrt[4x]{\sqrt[3x]{\sqrt[2x]{x}}}} & = \sqrt[x]{\sqrt[x^2]{\sqrt[x^3]{\sqrt[x^4]{x^5}}}}\\ {\left({\left({\left(x^{\tfrac{1}{2x}}\right)}^{\frac{1}{3x}}\right)}^{\frac{1}{4x}}\right)}^{\frac{1}{5x}} & = {\left({\left({\left({x^5}^{\tfrac{1}{x^4}}\right)}^{\frac{1}{x^3}}\right)}^{\frac{1}{x^2}}\right)}^{\frac{1}{x}}\\ x^{\tfrac{1}{120x^4}} & = x^{\tfrac{5}{x^{10}}}\\ \text{Equating the powers}:-\\ \dfrac{1}{120x^4} & = \dfrac{5}{x^{10}}\\ x^6 & = 120 × 5\\ x^6 & = 600\\ x & = \sqrt[6]{600}\\ \therefore a = \boxed{600} \end{aligned}

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