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Algebra Level 3

( x 4 x ) x 4 = x 4 x x 4 \Large {(\sqrt[4x]{x})}^{x^4} = \sqrt[x^4]{x^{4x}}

Find the real value of x x which satisfies the real equation above.

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

Note:- x ≠ 1 x \neq 1

This is one part of the set Fun with exponents


The answer is 16.

Ashish Menon by Ashish Menon , 20, India

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1 solution

Ashish Menon
Apr 25, 2016

( x 4 x ) x 4 = x 4 x x 4 ( x 1 4 x ) x 4 = ( x 4 x ) 1 x 4 x x 3 4 = x 4 x 3 Equating the powers : − x 3 4 = 4 x 3 x 6 = 16 x = 16 6 ∴ a = 16 \begin{aligned} {\left(\sqrt[4x]{x}\right)}^{x^4} & = \sqrt[x^4]{x^{4x}}\\ {\left(x^{\tfrac{1}{4x}}\right)}^{x^4} & = {\left(x^{4x}\right)}^{\frac{1}{x^4}}\\ x^{\tfrac{x^3}{4}} & = x^{\tfrac{4}{x^3}}\\ \text{Equating the powers}:-\\ \dfrac{x^3}{4} & = \dfrac{4}{x^3}\\ x^6 & = 16\\ x & = \sqrt[6]{16}\\ \therefore a & = \boxed{16} \end{aligned}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

I guessed -1 first then had to solve it 😂

Abhiram Rao - 5 years, 1 month ago

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Eee , fantastic XD

Ashish Menon - 5 years, 1 month ago

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