A difficult one

  • #1

Main Question or Discussion Point

express (2^1/2-1)^10 in the form k^1/2-(k-1)^1/2 where k is a positive integer.{the square roots need not be irrational}we can do this by binomial theorem but it is very tedious.is there a short & appropriate method to solve this problem?
 

Answers and Replies

  • #2
Tide
Science Advisor
Homework Helper
3,076
0
Just solve the equation

[tex]\sqrt{k} - \sqrt{k-1} = (\sqrt 2 - 1)^{10}[/tex]

algebraically for k. I get k = 11,309,769. This will be a mess unless you try something like

[tex]\sqrt k - \sqrt {k-1} = N[/tex]

from which

[tex]\sqrt k + \sqrt {k-1} = \frac {1}{N}[/tex]

leading to

[tex]2\sqrt k = N + \frac {1}{N}[/tex]

This is easy to solve for k and the solution can be simplified to what I showed above.
 

Related Threads on A difficult one

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
10
Views
5K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
4K
Top