To solve this equation, we can simplify it using the properties of radicals. Let's denote the first term as A and the second term as B:

A = √7 - 4√3
B = √7 + 4√3

Now, we can rewrite the given equation as:

A^x + B^x = 14

Next, let's recognize that A and B are conjugates of each other, which means that their product is a rational number:

A * B = (√7 - 4√3)(√7 + 4√3)
= 7 - 4(3)
= 7 - 12
= -5

Now, we raise this product to the power of x:

(A * B)^x = (-5)^x

Using the properties of exponents and the fact that A * B = -5, we can rewrite the original equation as:

(-5)^x = 14

As -5 raised to any real power will always be negative, it is impossible for it to be equal to 14. Therefore, there are no real solutions to the given equation.

In conclusion, the equation does not have any real solutions.