The Newton's Method is a method that approximates the solution of an equation.
It states that the zero of a function #P# can be approximated as #x_1=x_0-(P(x_0))/(P'(x_0))#, where #x_0# is an initial guess and #P'# is the derivative of #P# with respect to #x#. This can be used recursively to find more accurate solutions, using #x_1# as the second guess.
The derivative of #x^4-3# is #4x^3#. Then, our recursive equation is #x_1=x_0-(x_0^4-3)/(4x_0^3)#.
The question instructs us to start with #x_0=1#. Then, #x_1=1-(1^4-3)/(4*1^3)=3/2#.
Repeat the process with #x_0=3/2#: #x_1=3/2-((3/2)^4-3)/(4*(3/2)^3)=97/72~~1.35#. This is relatively close to one of the actual solutions: #root(4)(3)~~1.32#.
