Adv Alegbra HW Solutions 21

Adv Alegbra HW Solutions 21 - 1.46 True or false with...

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21 Solution. If a n = z = b n , then 1 = a n / b n = ( a / b ) n . Therefore, a / b is an n th root of unity, and so a / b = ζ k for some k ; that is, a = ζ k b . In particular, if b = w , then b = ζ k w . 1.45 (i) Find 8 + 15 i . Solution. The polar coordinates of ( 8 , 15 ) are ( 17 , 62 ) , and sin 31 . 515 and cos 31 . 857. Hence, 8 + 15 i 17 ( cos 62 + i sin 62 ) , and so 8 + 15 i 17 ( cos 31 + i sin 31 ) 3 . 533 + 2 . 123 i . (Of course, the other square root is the negative of this one.) (ii) Find all the fourth roots of 8 + 15 i . Solution. sin 15 . 5 . 267 and cos 15 . 5 . 967. Hence, 4 8 + 15 i = 4 17 ( cos 15 . 5 + i sin 15 . 5 ) 1 . 964 + 542 i . By Exercise 1.44, the other fourth roots are obtained by multiply- ing this one by i , 1, and i .
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Unformatted text preview: 1.46 True or false with reasons. (i) 6 | 2. Solution. False. (ii) 2 | 6. Solution. True. (iii) 6 | 0. Solution. True. (iv) | 6. Solution. False. (v) | 0. Solution. True. (vi) ( n , n + 1 ) = 1 for every natural number n . Solution. True. (vii) ( n , n + 2 ) = 2 for every natural number n . Solution. False. (viii) If b and m are positive integers, then b | m if and only if the last b-adic digit d of m is 0. Solution. True. (ix) 113 is a sum of distinct powers of 2. Solution. True....
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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