Answer
$L(x)=\frac{x}{12}-\frac{4}{3}$
Work Step by Step
Given: $f(x)=(x)^{\frac{1}{3}}$,
$L(x)=f(a)+f'(a)(x-a)$
$f'(x)=\frac{1}{3}{(x)^{\frac{-2}{3}}}$
$a=-8$
$f(-8)=(-8)^{\frac{1}{3}}=-2$
$f'(-8)=\frac{1}{3}{(-8)^{\frac{-2}{3}}}=\frac{1}{12}$
then $L(x)=-2+\frac{1}{12}(x+8)=\frac{x}{12}-\frac{4}{3}$
$L(x)=\frac{x}{12}-\frac{4}{3}$
Thus, the final answer is: $L(x)=\frac{x}{12}-\frac{4}{3}$
