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