Final answer:
To use Newton's method to find the fifth approximation to the root of the equation x⁵-x-1=0, start with x⁰=1 and iterate using the equation and its derivative.
Step-by-step explanation:
To find the fifth approximation to the root of the equation x⁵ -x-1=0 using Newton's method, we start with x⁰=1. First, we find the derivative of the equation, which is 5x⁴ -1. Then, we substitute x⁰=1 into the equation and the derivative to get x₁=1 - (1 - 1)/5 = 0.8.
Next, we repeat the process by substituting x₁ into the equation and the derivative, and continue iterating until we reach the fifth approximation.
The fifth approximation is x⁵ = 0.755.