A Radical Exponent

Algebra Level 1

$\large (x^2 +6)^{3+4{\sqrt{x}}-7}=1$

Find the integral solution of the above equation.

1 solution

Chung Kevin
Mar 5, 2016

Let $A = x^2 + 6$ and $B = 3+4\sqrt x - 7$ , so the equation becomes $A^B = 1$ .

If there exists a solution, then at least one of the following cases:

Case one : $A = 1$ .
Case two : $B = 0$ , $A\ne 0$ .
Case three : $A = -1$ , $B$ is an even number.

Solving these three cases shows that case two only yields a possible solution, with $3 + 4\sqrt x - 7 = 0 \Rightarrow x =\boxed1$ .