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Câu hỏi: Biết cho ${{9}^{x}}+{{9}^{-x}}=47.$ Khi đó giá trị của biểu thức $P=\dfrac{13+{{3}^{x}}+{{3}^{-x}}}{2-{{3}^{x}}-{{3}^{-x}}}$ bằng
A. $-\dfrac{5}{2}.$
B. 2.
C. $-4.$
D. $\dfrac{3}{2}.$
A. $-\dfrac{5}{2}.$
B. 2.
C. $-4.$
D. $\dfrac{3}{2}.$
Ta có ${{\left( {{3}^{x}}+{{3}^{-x}} \right)}^{2}}={{9}^{x}}+{{9}^{-x}}+2\Leftrightarrow {{\left( {{3}^{x}}+{{3}^{-x}} \right)}^{2}}=49\Leftrightarrow {{3}^{x}}+{{3}^{-x}}=7.$
Do vậy $P=\dfrac{13+{{3}^{x}}+{{3}^{-x}}}{2-{{3}^{x}}-{{3}^{-x}}}=\dfrac{13+{{3}^{x}}+{{3}^{-x}}}{2-\left( {{3}^{x}}+{{3}^{-x}} \right)}=\dfrac{13+7}{2-7}=-4.$
Vậy $P=\dfrac{13+{{3}^{x}}+{{3}^{-x}}}{2-{{3}^{x}}-{{3}^{-x}}}=-4.$
Do vậy $P=\dfrac{13+{{3}^{x}}+{{3}^{-x}}}{2-{{3}^{x}}-{{3}^{-x}}}=\dfrac{13+{{3}^{x}}+{{3}^{-x}}}{2-\left( {{3}^{x}}+{{3}^{-x}} \right)}=\dfrac{13+7}{2-7}=-4.$
Vậy $P=\dfrac{13+{{3}^{x}}+{{3}^{-x}}}{2-{{3}^{x}}-{{3}^{-x}}}=-4.$
Đáp án C.