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Câu hỏi: Giá trị của biểu thức $P={{\left( 7+4\sqrt{3} \right)}^{2022}}{{\left( 4\sqrt{3}-7 \right)}^{2021}}$ là
A. $P=-7+4\sqrt{3}$.
B. $P=-\left( 7+4\sqrt{3} \right)$.
C. $P=1$.
D. $P={{\left( 7+4\sqrt{3} \right)}^{2020}}$
A. $P=-7+4\sqrt{3}$.
B. $P=-\left( 7+4\sqrt{3} \right)$.
C. $P=1$.
D. $P={{\left( 7+4\sqrt{3} \right)}^{2020}}$
Ta có $\left( 7+4\sqrt{3} \right)\left( 4\sqrt{3}-7 \right)={{\left( 4\sqrt{3} \right)}^{2}}-{{7}^{2}}=-1$
$\Rightarrow P={{\left( 7+4\sqrt{3} \right)}^{2021}}{{\left( 4\sqrt{3}-7 \right)}^{2021}}={{\left[ \left( 7+4\sqrt{3} \right)\left( 4\sqrt{3}-7 \right) \right]}^{2021}}\left( 7+4\sqrt{3} \right)$
$={{\left( -1 \right)}^{2021}}\left( 7+4\sqrt{3} \right)=-\left( 7+4\sqrt{3} \right)$.
$\Rightarrow P={{\left( 7+4\sqrt{3} \right)}^{2021}}{{\left( 4\sqrt{3}-7 \right)}^{2021}}={{\left[ \left( 7+4\sqrt{3} \right)\left( 4\sqrt{3}-7 \right) \right]}^{2021}}\left( 7+4\sqrt{3} \right)$
$={{\left( -1 \right)}^{2021}}\left( 7+4\sqrt{3} \right)=-\left( 7+4\sqrt{3} \right)$.
Đáp án B.