Coursenotes_ECON301

First n1 means we are multiplying x by itself once

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Unformatted text preview: functional forms for n=1,2,3. First, n=1 means we are multiplying X by itself once which is simply X 4 Secondly, n=2 is the special case of the square function. This is X2 and produces a parabola. FIGURE 1: SQUARE FUNCTION 100 X -10 0 10 FIGURE 2: CUBIC FUNCTION If n=3 is the special case of the cubic function. This is X3 and produces an inflection point at 0. 64 -4 -2 8 0 -8 -64 2 4 X Now, what if n=0? There is not an intuitive way to evaluate the result of multiplying a number by itself zero times. It is generally accepted that X0 = 1. It is important to notice that the zeroth exponent is a definition and not a calculation. As an example, notice that we always have the following result: 10 20 15540 0 (-2.81)0 00 = = = = = = 1 1 1 1 1 1 Even negative numbers, or zero, have their zeroth exponent defined to be one. 5 Fractional Exponential Powers The concept of exponential powers can easily be extended to the case where the power is a fraction of the form 1/n or m/n. How do we define X to the power of 1/n? The exponential power X1/n means that if we repeatedly multiply X1/n by itself n times we get the original number X. X1/n X1/n X1/n X1/n ... X1/n = X { n times } or simply, [X1/n]n = X There is a close relationship between the integral exponential Xn and the fractional exponential X1/n. To see this, let's define a new variable Y = X1/n and then by raising Y to the exponent n (or multiplying Y by itself n times) we get the original variable X back. Yn = X1/n X1/n X1/n X1/n ... X1/n = X { n times } Let's look at a couple of common fractional exponents used in economics. First, X1/2, which means if we square it we'll get X back. This can also be thought of as the square root of X (X). FIGURE 3: SQUARE ROOT FUNCTION 2 1 0 1 4 X 6 Similarly, the exponential power X1/3 means that if we cube it we will get X back. This is also referred to as the cube root of X (3X). The graph is qualitatively the same as the square root function displayed above. Okay. Now let's calculate one of these things as...
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This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.

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