Answer
$$ \approx 1.9924716$$
Work Step by Step
$$\eqalign{
& \root 3 \of {7.91} \cr
& {\text{Using the result of Taylor polynomial of degree }}4{\text{ at }}x = 0{\text{ found in the exercise 8}} \cr
& f\left( x \right) = \root 3 \of {x + 8} {\text{ we found that}} \cr
& {P_4}\left( x \right) = 2 + \frac{1}{{12}}x - \frac{1}{{288}}{x^2} + \frac{5}{{20,736}}{x^3} - \frac{5}{{248,832}}{x^4} \cr
& {\text{To approximate }}\root 3 \of {7.91} ,{\text{ we must evaluate }}\sqrt { - 0.09 + 8} ,{\text{ then taking }}x = - 0.09 \cr
& {P_4}\left( { - 0.09} \right) = 2 + \frac{1}{{12}}\left( { - 0.09} \right) - \frac{1}{{288}}{\left( { - 0.09} \right)^2} + \frac{5}{{20,736}}{\left( { - 0.09} \right)^3} - \frac{5}{{248,832}}{\left( { - 0.09} \right)^4} \cr
& {P_4}\left( { - 0.09} \right) = 2 - 0.0075 - 0.000028125 - 0.000000175 - 0.000000001 \cr
& {P_4}\left( { - 0.09} \right) = 1.9924716 \cr
& {\text{Thus}}{\text{, }}\root 3 \of {7.91} \approx 1.9924716 \cr} $$
