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Question Number 96021 by SmNayon11 last updated on 29/May/20

$$\mathrm{if}\mathrm{p}\mathrm{and}\mathrm{q}\mathrm{are}\mathrm{two}\mathrm{complex}\mathrm{number}$$$$\mathrm{and}\mathrm{p}\times \mathrm{q}=\mathrm{m},\mathrm{m}\mathrm{is}\mathrm{a}\mathrm{real}\mathrm{number}.$$$$$$$$\mathrm{is}\mathrm{there}\mathrm{always}\mathrm{exists}\mathrm{a}{\mathrm{p}}^{\frac{1}{3}}\mathrm{and}{\mathrm{q}}^{\frac{1}{3}}$$$$(\mathrm{we}\mathrm{know}{\mathrm{p}}^{\frac{1}{3}}\mathrm{and}{\mathrm{q}}^{\frac{1}{3}}\mathrm{each}\mathrm{has}\mathrm{actually}$$$$3\mathrm{values})$$$$\mathrm{such}\mathrm{that}{\mathrm{p}}^{\frac{1}{3}}\times {\mathrm{q}}^{\frac{1}{3}}={\mathrm{m}}^{\frac{1}{3}}.\mathrm{where}{\mathrm{m}}^{\frac{1}{3}}$$$$\mathrm{is}\mathrm{real}.??\mathrm{how}\mathrm{to}\mathrm{prove}\mathrm{it}?$$$$$$

Commented by mr W last updated on 29/May/20

$$p,q\in \mathbb{C}$$$$m\in \mathbb{R}$$$$pq=m$$$$\Rightarrow {\left(pq\right)}^{\frac{1}{3}}=m^{\frac{1}{3}}$$$$\Rightarrow p^{\frac{1}{3}}{q}^{\frac{1}{3}}=m^{\frac{1}{3}}\notin \mathbb{R}$$$$example:$$$$p=i,q=-i\Rightarrow pq=1=m$$$$p^{\frac{1}{3}},q^{\frac{1}{3}}\in \mathbb{C}$$$$but{m}^{\frac{1}{3}}=thirdrootsofunit,\notin R$$

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