(81 * (-3)^x)^x=27

Question

*=multiplied

Answer 1

((3^4)*(-3)^x)^x=3^3;

(3^4x)*(-3)^x^2=3^3;

((-1)^x^2)(3^(4x+x^2)=3^3;

((-1)^x^2)(3^(4x+x^2-3)=1;

((-1)^x^2)(3^(x^2+4x-3)=1; x^2+4x-3=0 because 3^0=1

x^2+4x=3; x^2+4x+4=7; (x+2)^2=7, x=-2+sqrt(7); x^2=11-4sqrt(7) and (-1)^x^2 isn't definable.

Returning to the original problem, we can see that x must be an even value because the right side of the equation is positive; if x was odd, the right side would be negative. Let x=2n where n is an integer.

(3^8n*3^4n^2=3^3; 3^(8n+4n^2-3)=3^0; 4n^2+8n-3=0, n=(-8+sqrt(64+48))/8=(-8+4sqrt(7))/8=(-2+sqrt(7))/2. This is not an integer, so a solution cannot be found.

Suppose the equation had been: (81*3^-x)^x=27. Let's see if there is a solution:

(3^4x)*3^(-x^2)=3^3; 3^(4x-x^2-3)=3^0; x^2-4x+3=0=(x-3)(x-1), x=1 or 3.

CHECK: x=1: 81/3=27 OK; x=3: (81/27)^3=27 OK.

So it looks like the problem was incorrectly stated and should have been (81 * 3^(-x))^x=27.