Math Olympiad | Wonderful Radical Problem | 90% Failed to solve!

[@RyanLewis-Johnson-wq6xs]

Cbrt[77+20Sqrt[13]]+Cbrt[77-20Sqrt[13]]=7

[@RyanLewis-Johnson-wq6xs]

It’s in my head.

[@vijaymaths5483]

Great!!

[@RealQinnMalloryu4]

1^1^1+5^4+13^1+3^1^1^1 ➖ 5^4+13^1 1^2^2+1^1+3+1^2^2+^1^1 1^1+3 1^2 32 (x ➖ 3x+2).

[@RyanLewis-Johnson-wq6xs]

Cbrt[77+20Sqrt[3]]+Cbrt[77-20Sqrt[3]]=Cbrt[154+(Cbrt[265452957-68948820Sqrt[3]]/3)+3Cbrt[364133+94580Sqrt[3]]]