Calculate root(6)(-64)6√−64 means you have to find a real number xx such that x^6=-64x6=−64. Such number doesnt exist because if it were positive, then never will get a negative number as product, if it were negative, then
(-x)·(-x)·(-x)·(-x)·(-x)·(-x)=(−x)⋅(−x)⋅(−x)⋅(−x)⋅(−x)⋅(−x)= positive number (there are an even number of factors (6) and never will get -64−64)
In summary that root(6)(-64)6√−64 has no real solutions. There is no number xx such that x^6=-64x6=−64
But in complex set of numbers there are 6 solutions
First put -64−64 in polar form which is 64_18064180
Then the six solutions r_iri from i=0 to i=5 are
r_0=root(6)64_(180/6)=2_30r0=6√641806=230
r_1=root(6)64_((180+360)/6)=2_90r1=6√64180+3606=290
r_2=2_((180+720)/6)=2_150r2=2180+7206=2150
r_3=2_((180+1080)/6)=2_210r3=2180+10806=2210
r_4=2_270r4=2270
r_5=2_330r5=2330
Who are these numbers?
r_0=2(cos30+isin30)=sqrt3+ir0=2(cos30+isin30)=√3+i
r_1=2ir1=2i
r_2=-sqrt3+ir2=−√3+i
r_3=-sqrt3-ir3=−√3−i
r_4=-2ir4=−2i
r_5=sqrt3-ir5=√3−i
