SOLUTION: Please solve the following equation: {{{(sqrt(3))^(x+2)/27^x=3^(-x^2) }}} I am stumped as to how this can be solved. First, I did this: {{{(sqrt(3))^(x+2)/3^(3)^(x)=3^(-x^2) }

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Please solve the following equation: {{{(sqrt(3))^(x+2)/27^x=3^(-x^2) }}} I am stumped as to how this can be solved. First, I did this: {{{(sqrt(3))^(x+2)/3^(3)^(x)=3^(-x^2) }      Log On


   

Question 853665: Please solve the following equation:
%28sqrt%283%29%29%5E%28x%2B2%29%2F27%5Ex=3%5E%28-x%5E2%29+
I am stumped as to how this can be solved.
First, I did this:
%28sqrt%283%29%29%5E%28x%2B2%29%2F3%5E%283%29%5E%28x%29=3%5E%28-x%5E2%29+
Then:
%28sqrt%283%29%29%5E%28x%2B2%29=3%5E%28-x%5E2%29%2A+3%5E%283%29%5E%28x%29+
Found 2 solutions by Alan3354, josgarithmetic:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
%28sqrt%283%29%29%5E%28x%2B2%29%2F27%5Ex=3%5E%28-x%5E2%29+
= %28sqrt%283%29%29%5E%28x%2B2%29%2Fsqrt%283%29%5E%286x%29+=+3%5E%28-x%5E2%29+
= %28sqrt%283%29%29%5E%28x%2B2%29%2Fsqrt%283%29%5E%286x%29+=+sqrt%283%29%5E%28-2x%5E2%29+
= %28sqrt%283%29%29%5E%28-5x%2B2%29+=+sqrt%283%29%5E%28-2x%5E2%29+
-5x+%2B+2+=+-2x%5E2
2x%5E2+-+5x+%2B+2+=+0
(2x - 1)*(x - 2) = 0
x = 1/2
x = 2
You can check both answers.

Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Those two steps are good. Now the right member becomes 3%5E%283%5Ex-x%5E2%29.

Revising the equation then you have:
%28sqrt%283%29%29%5E%28x%2B2%29=3%5E%283%5Ex-x%5E2%29

You can make the left member be the base of 3, raised to a power:
3%5E%28%281%2F2%29%29%5E%28x%2B2%29=3%5E%283%5Ex-x%5E2%29
3%5E%28%28x%2B2%29%2F2%29=3%5E%283%5Ex-x%5E2%29----rendering cuts off the top; 3^((x+2)/2)=3^(3^x-x^2)

The left member is 3 raised to a power; the right member is 3 also raised to a power. Those powers, exponents, are equal.

%28x%2B2%29%2F2=3%5Ex-x%5E2
Still not easy to solve.
%28x%2B2%29%2F2%2Bx%5E2=3%5Ex
x%5E2%2B%281%2F2%29x%2B1=3%5Ex-----No ordinary algebra steps for this. Left and right members are separate functions which have one or more points (maybe) of intersection.