cross multiply to get:
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
Is there a more elegant solution than trial and error?
You have an excellent question!
In other words, your question was more like:
Find all solutions for n such that
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:
and find the derivative
and finally calculate a better approximation of x as
and proceed to calculate
which means that 0.786097643010236 is our (approximate) solution.
Hope I have answered your question.
algebra functions - how do i calculate this if f(x)= 3x+1 and g(x)=x+2 find (f+g...
algebra functions - how do i calculate this if f(x)= 3x+1 and g(x)=x+2 find (fg...
algebra with application - $4,000 at 10% for 20 years (a) Calculate the future ...
algebra - Calculate. 7[6+(8)^3]
Algebra - need help. The skier goes down the slope distance d and time t. The ...
Algebra - Calculate the value of |7.3| – |–2.1|
physics - 1. A student is dragging a 5 kg box along the floor. He pulls on the ...
chemistry - When 10.0 mL of 0.012 M Pb(No3)2 is mixed with 10.0 mL of 0.030M KI...
college physics - Consider two vectors A = 4i - 4j and B = -i - 8j . (a) ...
college algebra - Slopes are all around you! Take a look around where you are ...