Coursenotes_ECON301

What is 2532 2532 2512 2512 2512 555 125 how do

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Unformatted text preview: an example. What is 253/2? 253/2 = 251/2 251/2 251/2 =555 = 125 How do we get back to 25 from 125? We want to raise 125 to the exponent 2/3. To see this, consider... 1252/3 = [125 125]1/3 = [15625]1/3 = [25 25 25]1/3 = 25 So we have the result that 253/2 = 125 and 1252/3 = 25. EXPONENTIAL POWER ARITHMETIC RULES Exponential powers provide a convenient way for us to perform multiplication and division of numbers by performing simple addition and subtraction on their exponential powers. Multiplication: To multiply two exponential powers Xm and Xn of a given number X, we can simply add the powers m and n together. Xm Xn = Xm+n This can be seen as follows, Xm Xn = (X X X X ... X) (X X X X ... X) { m times } { n times } = (X X X X ... X) { m+n times } = Xm+n 7 Example, X3 X4 = (X X X) (X X X X) =XXXXXXX = X7 = X3+4 Division: To divide two exponential powers Xm and Xn of a given positive number X, we subtract the powers m and n. Xm / Xn = Xm-n This can be seen as follows: Xm / Xn = (X X X X ... X) / (X X X X ... X) { m times } { n times } = (X X X X ... X) { m-n times } = Xm-n Example, X6 / X4 = (X X X X X X) / (X X X X) =XXXXXX XXXX = X2 = X(6-4) Okay, but what if m=0? If m=0, then m - n = -n. We'll use the fact that the power 0 of a number is defined to be 1, to get... X-n = X0-n = X0 Xn =1 Xn 8 That is, raising a number to a negative power is equivalent to taking its reciprocal. For example, X-2 = 1 X2 X-6 = 1 X6 X-1/2 = 1 X1/2 =1 X X-2/3 = 1 X2/3 Finally, to take the exponential power m of an exponential power Xn, we multiply the powers m and n together. [Xn]m = Xn Xn Xn Xn Xn ... Xn ( m times ) = (X X ... X) (X X ... X) ... (X X ... X) ( n times ) ( n times ) ( n times ) { m times } =XX...X (nm times) = Xnm For example, [X3]2 = X3 X3 = (X X X) (X X X) =XXXXXX = X6 = X32 = Xnm 9 Let's summarize, PROPERTIES OF EXPONENTIAL POWERS X0 = 1 Xn = X X X X ... X ( n times ) X-n = 1 Xn Xm Xn = Xm+n Xm / Xn = Xm-n [Xn]m = Xnm Exponential powers are used widely in economic functional forms. Some examples include, SQUARE ROOT UTILITY U(X,Y) = XY = X...
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