Question

simplify and prove your answer : ³ √8+3√21. + ³√8-3√21​

Answer 1

this is the solution to a depressed cubic that’s easy to reverse engineer. Let’s let

a=8+321−−√−−−−−−−−√3,b=8−321−−√−−−−−−−−√3

so

ab=64−9(21)−−−−−−−−√3=−125−−−−√3=−5

x=a+b

x3=(a+b)3=a3+3a2b+3ab2+b3

x3=a3+b3+3ab(a+b)=a3+b3+3abx

x3=8+321−−√+8−321−−√+3(−5)x

x3+15x−16=0

with single real solution  x=1