hw5solutions - Math 521 - Advanced Calculus I Instructor:...

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Unformatted text preview: Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: February 1, 2010 Assignment 5 1. Fix b > 1. (a) If m,n,p,q are integers, n > 0, q > 0, and r = m/n = p/q , prove that ( b m ) 1 /n = ( b p ) 1 /q . Hence it makes sense to define b r = ( b m ) 1 /n . (b) Prove that b r + s = b r b s if r and s are rational. (c) If x is real, define B ( x ) to be the set of all numbers b t , where t is rational and t x . Prove that b r = sup B ( r ) when r is rational. Hence it makes sense to define b x = sup B ( x ) for every real x . (d) Prove that b x + y = b x b y for all real x and y . (a) If m/n = p/q , we have mq = np . By definition, we have y = ( b m ) 1 /n satisfies y n = b m . Similarly, we have z = ( b p ) 1 /q satisfies z q = b p . Then, y np = b mp = z qm . Since np = qm N , we can take this root of both sides. By uniqueness of n th roots, we have that y = z . (b) Let r,s Q . By finding a common denominator, we can write r = m/n and s = l/n . Then, b r + s = b ( m + l ) /n = ( b ( m + l ) ) 1 /n = ( b m b l ) 1 /n ....
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hw5solutions - Math 521 - Advanced Calculus I Instructor:...

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