Newtons Method and Finding the 5th root

[1irishman]

Homework Statement

Find the 5th root of 36 accurate to four decimal places

Homework Equations

xn+1 = xn - f(xn)/f'(xn)

The Attempt at a Solution

First I attempted to write the fifth root of 36 in exponential form as show below:

Let the 5th root of 36 = x
Let f(x) = x^1/5 - 36
So, f'(x) = 1/5x^-4/5 Is this right so far?

 

[LCKurtz]

Homework Statement

Find the 5th root of 36 accurate to four decimal places

Homework Equations

xn+1 = xn - f(xn)/f'(xn)

The Attempt at a Solution

First I attempted to write the fifth root of 36 in exponential form as show below:

Let the 5th root of 36 = x
Let f(x) = x^1/5 - 36
So, f'(x) = 1/5x^-4/5 Is this right so far?

No. You need to solve x⁵ - 36 = 0.

 

[1irishman]

No. You need to solve x⁵ - 36 = 0.

oh okay, but i am confused because i thought that the fifth root of a number in exponent
form was that number raised to the 1/5. I don't understand how it is x raised to the fifth.

 

[LCKurtz]

No. You need to solve x⁵ - 36 = 0.

oh okay, but i am confused because i thought that the fifth root of a number in exponent
form was that number raised to the 1/5. I don't understand how it is x raised to the fifth.

If you solved that equation by radicals wouldn't you get x=36^{\frac 1 5}? And isn't that what you are asked to find? So you want the root of that equation.

 

[1irishman]

oh okay. So then it is like below?

f(2) = 2^5 - 36
= - 4
and

f'(2) = 5(2^4)
= 80

 

[LCKurtz]

oh okay. So then it is like below?

f(2) = 2^5 - 36
= - 4
and

f'(2) = 5(2^4)
= 80

No. Don't you have any worked examples in your text? You mentioned above the recursion formula for x_n+1. What do you get for that? You need to use it.

 

[1irishman]

it's okay, it says that in the text...got it from here. thx.