Friday, 31 March 2017

algebra precalculus - Why is sqrt[3]18+5sqrt13+sqrt[3]185sqrt13=3?





How to show that 318+513+318513=3?




This equality comes from solving t315t4=0

using Cardanos fomula and knowing the solution t1=4.



I have attempted multiplying the whole thing with (318+513)2(318513)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+b13)3=(18+513) for a,bQ



Expanding the LHS gives,




(a3+39ab218)+13(3a2b+13b35)=0

,



From this we get,



{a3+39ab218=03a2b+13b35=0



Solving the system give a=32 and b=12



Therefore




3(18+513)=32+1213



Similarly,



3(18513)=321213



Hence the sum is 3.


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