302 HW 5 - a 2 Since 5 is prime 5 must divide a so a = 5c...

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M302 HOMEWORK #5 Dr. Schurle Assignment: page 118, #3, 6, 8, 10, 12, 30 Grade the following for 2 points each, then give all or part of the remaining two points depending on how much work they've done on the other problems. #3. 1 2 5 2 = 6 2 = 3 1 2 2 3 = 3 6 4 6 = 1 6 1 2 × 6 5 = 6 10 = 3 5 1 2 2 3 = 1 2 × 3 2 = 3 4 The last one becomes 3 over 2/3, which is 3 times 3/2, which is 9/2. #8. 4/9 is a quotient of two integers and so is rational. 1.75 is a terminating decimal, so it is rational. 20 3 5 = 4 3 = 2 3 , which is rational. If 2 14 were rational, then 2 would be 14 times a rational, which would be rational, and this is false. So 2 14 must be irrational. 3.14149 is a terminating decimal, so it is rational.
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#10. Suppose 5 is rational, so that 5 = a b , where a and b are integers and the fraction is completely reduced. Then 5 = a 2 b 2 so 5b 2 = a 2 and 5 divides
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Unformatted text preview: a 2 . Since 5 is prime, 5 must divide a, so a = 5c and then we get 5b 2 = 5c 2 = 25c 2 . Divide by 5 to get b 2 = 5c 2 . Just as with a, we see that now 5 must divide b, so the fraction a/ b is NOT completely reduced. This is big trouble, so 5 cannot be rational. #30. Suppose 3 2 is rational, so that 3 2 = a b , where a and b are integers and the fraction is completely reduced. Cube both sides to get 2 = a 3 b 3 so 2b 3 = a 3 and 2 divides a 3 . Since 2 is prime, 2 must divide a, so a = 2c and then we get 2b 3 = 2c 3 = 8c 3 . Divide by 2 to get b 3 = 4c 3 . Just as with a, we see that now 2 must divide b , so the fraction a/b is NOT completely reduced. This is big trouble, so 3 2 cannot be rational....
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