If the real valued function f(x)=(a^x-1)/(x^n≤ft(a^x+1)) is even then…

$f(x)=\frac{a^x-1}{x^n\left(a^x+1\right)}$

$f(x)$ → even

$⇒f(-x)=f(x)$

so $f(-x)=\frac{a^{-x}-1}{(-x)^n(a^{-x}+1)}=\frac{1-a^x}{(-x)^n(1+a^x)}$

so $f(-x)=-1×(-x)^n×\frac{a^x-1}{a^x+1}$

$⇒-(-x)^n=(x)^n$

$⇒n=1/3$