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if-p-and-q-are-two-complex-number-and-p-q-m-m-is-a-real-number-is-there-always-exists-a-p-1-3-and-q-1-3-we-know-p-1-3-and-q-1-3-each-has-actually-3-values-such-that-p-1-3-q-1-3




Question Number 96021 by SmNayon11 last updated on 29/May/20
if p and q are two complex number  and p×q=m  ,m is a real number .    is there always exists a  p^(1/3)  and q^(1/3)   (we know p^(1/3)  and q^(1/3) each has actually   3 values)  such that p^(1/3) ×q^(1/3) =m^(1/3) .where m^(1/3)   is real .??  how to prove it?
ifpandqaretwocomplexnumberandp×q=m,misarealnumber.istherealwaysexistsap13andq13(weknowp13andq13eachhasactually3values)suchthatp13×q13=m13.wherem13isreal.??howtoproveit?
Commented by mr W last updated on 29/May/20
p,q∈C  m∈R  pq=m  ⇒(pq)^(1/3) =m^(1/3)   ⇒p^(1/3) q^(1/3) =m^(1/3) ∉R  example:  p=i, q=−i ⇒pq=1=m  p^(1/3) ,q^(1/3) ∈C  but m^(1/3) =third roots of unit, ∉R
p,qCmRpq=m(pq)13=m13p13q13=m13Rexample:p=i,q=ipq=1=mp13,q13Cbutm13=thirdrootsofunit,R

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