Newton's Method (problem solving)

In summary, the conversation is discussing finding the sixth root of 2 using Newton's Method. The book presents the problem as finding (2)^1\6 correct to eight decimal places and explains that this is equivalent to finding the positive root of the equation x^6 - 2 = 0. The person asking the question is looking for a brief explanation of why this is equivalent and someone provides an example using x^6 - 2 = 0 to show that the root of this equation is equal to 2^(1/6). They then mention using Newton's method to find the root and thank the person for their explanation. They also mention that there are six roots to the equation x^6 - 2 = 0
  • #1
mateomy
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Homework Statement



Its not an actual problem, just a question. We're being lectured on Newton's Method, which I understand. But there's a section in the book (and related problems later in the chapter), that ask for you to find (2)^1\6 correct to eight decimal places. It goes on to say that it is equivalent to finding the positive root of the equation and breaks it down to x^6 - 2 = 0. I don't remember finding roots from my previous classes. I know it probably has something to do with variation in sign (Descartes?). Again, this isn't a problem, but can somebody just briefly explain to me the "this is equivalent to finding the positive root" part?

THANKS!
 
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  • #2
Let's say that 2^(1/6) = x (which is true, because we define x that way. Then...

2^(1/6)=x
2=x^6
0=x^6-2

So you can see that the root of x^6-2 would be 2^(1/6). Then you use Newton's method to find the root.
 
  • #3
Got it. Thanks for taking the time to explain.
 
  • #4
The equation x6 - 2 = 0 (or equivalently, x6 = 2) has six roots. Four of them are complex and two of them are real, with one positive and one negative. Each of the roots of this equation is a sixth root of 2, or [tex]\sqrt[6]{2}[/tex]
 

1. What is Newton's Method?

Newton's Method is a problem-solving algorithm used to find the roots of a function. It is based on the idea of using an initial guess to iteratively improve the approximation to the root of the function.

2. How does Newton's Method work?

The method starts with an initial guess for the root of the function. Then, using the derivative of the function, it calculates the slope of the tangent line at that point. The intersection of the tangent line with the x-axis gives a new approximation for the root. This process is repeated until the desired level of accuracy is achieved.

3. What types of problems can be solved using Newton's Method?

Newton's Method is commonly used to find the roots of a function, but it can also be applied to solve optimization problems such as finding the maximum or minimum of a function.

4. What are the limitations of Newton's Method?

Newton's Method may not always converge to a solution, especially if the initial guess is far from the actual root or if the function has multiple roots. Additionally, the method may converge to a local minimum or maximum instead of the global minimum or maximum.

5. Are there any alternatives to Newton's Method?

Yes, there are other methods for solving equations and optimization problems such as the Bisection Method, Secant Method, and the Quasi-Newton Method. Each method has its own advantages and limitations, and the choice of method depends on the specific problem at hand.

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