Answer
$f^{-1}(x)=\sqrt[5]{\sqrt[7]{x}+6}$
Work Step by Step
$f(x)=(x^{5}-6)^{7}$
Rewrite this expression as $y=(x^{5}-6)^{7}$ and solve for $x$:
$y=(x^{5}-6)^{7}$
Take the seventh root of both sides:
$\sqrt[7]{y}=\sqrt[7]{(x^{5}-6)^{7}}$
$\sqrt[7]{y}=x^{5}-6$
Take $-6$ to the left side:
$\sqrt[7]{y}+6=x^{5}$
$x^{5}=\sqrt[7]{y}+6$
Take the fifth root of both sides:
$\sqrt[5]{x^{5}}=\sqrt[5]{\sqrt[7]{y}+6}$
$x=\sqrt[5]{\sqrt[7]{y}+6}$
Interchange $x$ and $y$:
$y=\sqrt[5]{\sqrt[7]{x}+6}$
The inverse of the original function is $f^{-1}(x)=\sqrt[5]{\sqrt[7]{x}+6}$