Question 1209711
Here's how to approach this problem:

1. **Consider x = 1:**

If x = 1, the equation becomes:

[1 - (1/1)]^(1/1) + [1 - (1/1)]^(1/1) = 1

[1 - 1]^1 + [1 - 1]^1 = 1

0^1 + 0^1 = 1

0 + 0 = 1

0 = 1

This is not true, so x = 1 is not a solution.

2. **Rewrite the equation:**

Let's rewrite the given equation:

(x - 1)/x ]^(1/x) + [(x - 1)/x]^(1/x) = x

2 * [(x - 1)/x]^(1/x) = x

3. **Consider the case where x = 2:**

2 * [(2 - 1)/2]^(1/2) = 2

2 * (1/2)^(1/2) = 2

2 * (1/√2) = 2

√2 = 2

This is not true, so x=2 is not a solution.

4. **Consider the case where x = 4:**

2 * [(4 - 1)/4]^(1/4) = 4

2 * (3/4)^(1/4) = 4

(3/4)^(1/4) = 2

Raising both sides to the power of 4:

3/4 = 2^4

3/4 = 16

This is not true, so x=4 is not a solution.

5. **A more general approach is difficult:** This equation is a transcendental equation (it mixes polynomials and non-algebraic functions).  There's no simple algebraic method to solve it.  Numerical methods (like the Newton-Raphson method) or a computer algebra system are usually required.

6. **Graphical Approach:**  The best way to find a solution is to graph both sides of the equation, y = 2 * [(x - 1)/x]^(1/x) and y = x, and look for the intersection point.  This will give you an approximate solution.

7. **Numerical Solution:**  Using a numerical solver (like Wolfram Alpha or a calculator with a solve function), we can find an approximate solution. It will be a value greater than 1.

It is difficult to find a closed-form solution. The most practical approach is to use a numerical solver or a graphing calculator to approximate the solution.