Chapter 4.8, Problem 39E

Estimating roots The values of various roots can be approximated using Newton’s method. For example, to approximate the value of $\sqrt[3]{10}.$ we let $x=\sqrt[3]{10}$ and cube both sides of the equation to obtain x³ = 10. or x³ – 10 = 0. Therefore, $\sqrt[3]{10}$ is a root of p(x) = x³ – 10, which we can approximate by applying Newton’s method. Approximate each value of r by first finding a polynomial with integer coefficients that has a root r. Use an appropriate value of xo and stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding.

39. $r=7^{1/4}$