You can put this solution on YOUR website! 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.
