{Use of Tech} Estimating roots The values of various roots can be approximated using Newton’s method. For example, to approximate the value of ³√10, we let x = ³√10 and cube both sides of the equation to obtain x³ = 10, or x³ - 10 = 0. Therefore, 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 x₀ and stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding.


r = 7¹/⁴

Verified step by step guidance

1. Identify the polynomial that has the root r = 7^(1/4). This can be expressed as p(x) = x^4 - 7, since 7^(1/4) is a root of this polynomial.
2. Choose an initial guess x₀ for the root. A reasonable starting point could be x₀ = 2, since 2^4 = 16 is greater than 7, and 1 is less than 7.
3. Apply Newton's method using the formula x_{n+1} = x_n - rac{p(x_n)}{p'(x_n)}, where p'(x) is the derivative of p(x). Calculate p'(x) = 4x^3.
4. Calculate the first iteration using your initial guess x₀. Substitute x₀ into the formula to find x₁, and then repeat this process to find x₂.
5. Continue iterating until the difference between two successive approximations is less than 0.00001, ensuring that they agree to five decimal places.