Simple Operations

• 1). Define the exponential function C^x, if x<0, to be the reciprocal of C^(-x). For example, 2^(-2) equals 1/(2^2)=1/4.
  
  
• 2). Define the exponential function C^x to be 1 if x=0. This makes sense because, for example, 2^3 / 2^3 = 1, and 2^3 / 2^3 = 2^(3-3). Therefore, it makes sense to define 2^0 as 1.
  
  
• 3). Define the exponential function C^x when x is not integer but is rational as demonstrated in this example. Denote 2^(3/4) with the letter E, as a placeholder, since it is the unknown but you still want to be able manipulate it algebraically. E^4=2^3=8. So solve for the number E that, when raised to the fourth power, equals 8. You may need to use a trial-and-error method from the field of numerical analysis to find that number E though. See the next section on how to do so.
  
  

Approximations

• 1). Determine the exponential function C^x when x is irrational by approximation, first by rounding x to a certain number of digits.
  
  For example, suppose you want to determine 2^x at x=√2. The square root of 2 is 1.4142.... Round it to 1.414.
  
  
• 2). Divide the rounded exponent by a factor of 10 to get rid of the decimal point.
  
  Continuing with the above example, 2^1.414 becomes 2^(1414/1000).
  
  
• 3). Set the exponential function with the letter E. Raise both sides to a power equal to the denominator of the exponent. Move both pieces to the same side of the equals sign. Define this is a new function. Keep in mind that you want to solve for E.
  
  Continuing with the above example, 2^(1414/1000) = E. Therefore, 2^1414 = E^1000. Move it all to one side and making it a new function: f(E) = 2^1414 - E^1000 = 0
  
  
• 4). Guess at one value of E, denoting it E_1. Plug it into f(E) and see what its sign is. Then find a nearby value of E that gives the opposite sign. Call it E_2. Don't actually solve for it using a calculator--or even Excel. They don't register exponents that high. You'll need to program a computer to do this.
  
  
• 5). Take the average of E_1 and E_2. Call it E_3. Find the sign of f(E_3). Substitute E_3 for which of E_1 and E_2 produces the same sign for f(E).
  
  
• 6). Repeat Step 5 until you get E_n as close to the real E as you want. In the field of numerical analysis, this iteration procedure is called the bisection method.