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Curious Cubic Calculation? (Posted on 2004-10-13) Difficulty: 2 of 5
Show that α=√(7+5√2)+√(7-5√2) is an integer. No calculators or computers allowed!

See The Solution Submitted by Old Original Oskar!    
Rating: 3.8333 (6 votes)

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Solution Algebraically | Comment 9 of 14 |
If (a+b√2)³=7+5√2 then (a-b√2)³=7-5√2, so if we can find a and b, the answer will be 2a.

Algebraically, (a+b√2)³= a³+3a²b√2+6ab²+2b³√2, so we need a³+6ab²=7 and 3a²b+2b³=5.

From the second, a²=(5-2b³)/(3b), so from the first, after some simplifying, we get a[(16b³+5)/(3b)]=7.

As we want integer solutions, either a=1 or a=7; the first works out, with b=1, so the final answer is 2.
  Posted by e.g. on 2004-10-13 16:24:58

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