Algebra

posted by .

How do I calculate 4^n=1/(2(n-n^2))? Thanks.

  • Algebra -

    cross multiply to get:
    2(n-n^2)4^n=1
    (n-n^2)4^n=1/2
    Since there is no more division and the right-hand side is less than 1, we can start searching for n where n<1, say 1/2.
    Try n=1/2 and see if the equation is satisfied.
    Hint: one of the laws of exponents tells us that
    X^(1/2)=√(X).

  • Algebra -

    Is there a more elegant solution than trial and error?

  • Algebra -

    You have an excellent question!

    In other words, your question was more like:
    Find all solutions for n such that
    4^n=1/(2(n-n^2))

    I do not see an explicit solution to the equation. Perhaps someone else can find one.

    An explicit solution is expressed as
    n=expression where expression does not contain n.

    Lack of an explicit solution, I proceed as follows:

    1. first bound the solution.
    We can conclude that for n outside of [-1,1], we cannot have the right hand side equal to -1.

    2. Find approximate solutions by graphing or otherwise. By graphing, there are two solutions, at 0.5 and 0.8, approximately.

    3. proceed to refine the solutions by iterations (a glamorous name for trial and error).

    Here are the details of iteration using Newton's method:

    For the case of n=0.5, it is exact and so requires no further iteration.

    For the case of n=0.8, we can refine the solution by setting up Newton's iteration equation:

    let f(x)=4^x-1/(2*(x-x^2))
    and find the derivative
    f'(x)=log(4)4^x+(1-2x)/(2(x-x^2)^2)
    and finally calculate a better approximation of x as
    f1(x)=x-f(x)/f'(x)
    and proceed to calculate
    x0=0.8,
    x1=f1(x0)=0.787551458195669
    x2=f1(x1)=0.786114835166431
    x3=f1(x2)=0.786097645437988
    x4=f1(x3)=0.786097643010236
    x5=f1(x4)=0.786097643010236
    which means that 0.786097643010236 is our (approximate) solution.

    Hope I have answered your question.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Algebra

    Calculate the value of |7.3| – |–2.1|
  2. statistics

    1. Use the following information about a group of college students: Fifty percent are taking Algebra. Sixty percent are taking Biology. Thirty percent are taking both Biology and Algebra. A student from the group is selected at random, …
  3. algebra with application

    $4,000 at 10% for 20 years (a) Calculate the future amount if the money were invested at simple interest. Calculate the future amount if the money were invested with annual compounding. (Round your answer to the nearest cent.)
  4. algebra

    Calculate. 7[6+(8)^3]
  5. Algebra

    need help. The skier goes down the slope distance d and time t. The relationship between d and t can be represented in the following form d = at = bt^2 where a nd b are constants. If given d=11 t=2 and d=28 t=4 how to calculate values …
  6. algebra functions

    how do i calculate this if f(x)= 3x+1 and g(x)=x+2 find (f+g)(x) and calculate (f+g)(1) a little confused can some one please help me thank you
  7. algebra functions

    how do i calculate this if f(x)= 3x+1 and g(x)=x+2 find (fg)(x)and calculate (fg)(2) is this multiplication and if so what do i do first any help would be appreciated
  8. Statistics for Business

    Consider the following sample of data: 11.2, 28.4, 32.6, 17, 13.5, 2.2, 38, 41.4, 1.9, 14.4, 21.8, 23.8 a) Calculate the max, min, Q1, median, and Q3 b) Calculate the average c) Calculate the standard deviation d) Calculate the variance
  9. Algebra

    Define #N by the formula #N=.5(N)+1. Calculate #(#(#50)). Thank you!
  10. algebra

    Calculate: -2/3 -4

More Similar Questions