4 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c ° W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 4 INDICES AND LOGARITHMS 4.1. Indices Given any non-zero real number a R and any natural number k N , we often write a k = a × ... × a | {z } k . (1) This deFnition can be extended to all integers k Z by writing a 0 = 1 (2) and a k = 1 a k = 1 a × × a | {z } k (3) whenever k is a negative integer, noting that k N in this case. It is not too difficult to establish the following. LAWS OF INTEGER INDICES. Suppose that a, b R are non-zero. Then for every m, n Z , we have (a) a m a n = a m + n ; (b) a m a n = a m n ; (c) ( a m ) n = a mn ; and (d) ( ab ) m = a m b m . This chapter was written at Macquarie University in 1999.
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4–2 W W L Chen and X T Duong : Elementary Mathematics We now further extend the deFnition of a k to all rational numbers k Q . To do this, we Frst of all need to discuss the q -th roots of a positive real number a , where q N . This is an extension of the idea of square roots discussed in Section 1.2. We recall the following deFnition, slightly modiFed here. Definition. Suppose that a R is positive. We say that x> 0 is the positive square root of a if x 2 = a . In this case, we write x = a . We now make the following natural extension. Definition. Suppose that a R is positive and q N . We say that 0 is the positive q -th root of a if x q = a . In this case, we write x = q a = a 1 /q . Recall now that every rational number k Q can be written in the form k = p/q , where p Z and q N . We may, if we wish, assume that p/q is in lowest terms, where p and q have no common factors. ±or any positive real number a R , we can now deFne a k by writing a k = a p/q =( a 1 /q ) p a p ) 1 /q . (4) In other words, we Frst of all calculate the positive q -th root of a , and then take the p -th power of this q -th root. Alternatively, we can Frst of all take the p -th power of a , and then calculate the positive q -th root of this p -th power. We can establish the following generalization of the Laws of integer indices. LAWS OF INDICES. Suppose that a, b R are positive. Then for every m, n Q , we have (a) a m a n = a m + n ; (b) a m a n = a m n ; (c) ( a m ) n = a mn ; and (d) ( ab ) m = a m b m . Remarks. (1) Note that we have to make the restriction that the real numbers a and b are positive. If a = 0, then a k is clearly not deFned when k is a negative integer. If a< 0, then we will have problems taking square roots. (2) It is possible to deFne cube roots of a negative real number a . It is a real number x satisfying the requirement x 3 = a . Note that x< 0 in this case. A similar argument applies to q -th roots when q N is odd. However, if q N is even, then x q 0 for every x R , and so x q 6 = a for any negative a R . Hence a negative real number does not have real
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4 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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