How to show that 3√18+5√13+3√18−5√13=3?
This equality comes from solving t3−15t−4=0
using Cardanos fomula and knowing the solution t1=4.
I have attempted multiplying the whole thing with (3√18+5√13)2−(3√18−5√13)2, but no success. Then I have solved for one cubic root and put all to the third power. Also no success.
Answer
Let (a+b√13)3=(18+5√13) for a,b∈Q
Expanding the LHS gives,
(a3+39ab2−18)+√13(3a2b+13b3−5)=0
,
From this we get,
{a3+39ab2−18=03a2b+13b3−5=0
Solving the system give a=32 and b=12
Therefore
3√(18+5√13)=32+12√13
Similarly,
3√(18−5√13)=32−12√13
Hence the sum is 3.
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