If P^0.223 =1 + P^0.123. Solve for P. Note:only a genius can solve this. Best of luck
No genius required. Just use your favorite numerical method to find the root:
x^.223 - x^.123 - 1 = 0
x = 92.6740
Newton-Raphson is a likely one for this, since the curve is nice and smooth
Solve without newton-raphson method
To solve the equation P^0.223 = 1 + P^0.123 for P, we will use a trial and error method. Here's how you can approach this problem:
1. Start by assigning a value to P and substitute it into the equation.
For example, let's assume P = 1.
P^0.223 = 1^0.223 = 1
Similarly, P^0.123 = 1^0.123 = 1
2. Substitute these values into the equation and check if they satisfy the equation.
1 = 1 + 1
1 = 2
Since this equation is not true, we need to try another value for P.
3. Repeat the process for several values of P until you find a value that satisfies the equation.
Let's try P = 2.
P^0.223 = 2^0.223 ≈ 1.2909
P^0.123 = 2^0.123 ≈ 1.1267
Substitute these values into the equation:
1.2909 ≠ 1 + 1.1267
Since this is not true, let's try another value for P.
4. Continue this process by trying different values of P until we find a value that satisfies the equation.
Let's try P = 3.
P^0.223 = 3^0.223 ≈ 1.3822
P^0.123 = 3^0.123 ≈ 1.145
Substitute these values into the equation:
1.3822 ≠ 1 + 1.145
This is also not true, so we'll continue trying other values.
5. Proceed with this trial and error process, increasing the value of P each time. Eventually, you will find a value of P that satisfies the equation.
Since this method requires trying different values and checking if they satisfy the equation, it may take a few attempts before finding the correct value of P. This problem does not have a direct algebraic solution, so it relies on numerical approximation methods. Therefore, finding the exact value without the use of a computer program may be challenging.