Rút gọn biểu thức \(P=(\sqrt[3]{9+\sqrt{80}})^{2020} \cdot(3-\sqrt[3]{9+\sqrt{80}})^{2021} .\)
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Lời giải:
Báo sai\(\begin{aligned} &\text { Đặt } x=\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}} \text { ta có } \\ &x^{3}=9+\sqrt{80}+3 \cdot(\sqrt[3]{9+\sqrt{80}})^{2} \cdot \sqrt[3]{9-\sqrt{80}}+3 . \sqrt[3]{9+\sqrt{80}} \cdot(\sqrt[3]{9-\sqrt{80}})^{2}+9-\sqrt{80} \\ &=18+3 \cdot \sqrt[3]{9+\sqrt{80}} \cdot \sqrt[3]{9-\sqrt{80}}(\cdot \sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}) \\ &=18+3 x \cdot \sqrt[3]{9+\sqrt{80}} \cdot \sqrt[3]{9-\sqrt{80}}=18+3 x \Rightarrow x=3 \Rightarrow 3-\sqrt[3]{9+\sqrt{80}}=\sqrt[3]{9-\sqrt{80}} \end{aligned}\)
Khi đó:
\(\begin{aligned} &P=(\sqrt[3]{9+\sqrt{80}})^{2020} \cdot(3-\sqrt[3]{9+\sqrt{80}})^{2021}=(\sqrt[3]{9+\sqrt{80}})^{2020} \cdot(\sqrt[3]{9-\sqrt{80}})^{2021} \\ &=(\sqrt[3]{9+\sqrt{80}} \cdot \sqrt[3]{9-\sqrt{80}})^{2020} \cdot \sqrt[3]{9-\sqrt{80}}=(\sqrt[3]{1})^{2020} \cdot \sqrt[3]{9-\sqrt{80}}=\sqrt[3]{9-\sqrt{80}} \end{aligned}\)