Giải bài 10 trang 78 SGK giải tích nâng cao 12
Chứng minh
a) \(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}=2\); b) \(\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}=3\)
a) Ta có:
\(\begin{align} & \sqrt{4+2\sqrt{3}}=\sqrt{{3}+2\sqrt{3}+1} \\ & =\sqrt{{{\left( \sqrt{3}+1 \right)}^{2}}} \\ & =\sqrt{3}+1 \end{align} \)
Tương tự \(\sqrt{4-2\sqrt{3}}=\sqrt{3}-1\)
Suy ra \(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}=2\) (đpcm)
b) Ta có
\(\begin{align} \sqrt[3]{9+\sqrt{80}}& =\sqrt[3]{9+4\sqrt{5}} \\ & =\sqrt[3]{5+4\sqrt{5}+4} \\ & =\sqrt[3]{{{\left( \sqrt{5}+2 \right)}^{2}}} \end{align}\)
Tương tự \(\sqrt[3]{9-\sqrt{80}}=\sqrt[3]{{{\left( \sqrt{5}-2 \right)}^{2}}}\)
Đặt \(\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}=t\)
Nhắc lại: \({{\left( x+y \right)}^{3}}={{x}^{3}}+{{y}^{3}}+3xy\left( x+y \right)\)
Ta có
\({{\left( t \right)}^{3}}=9+\sqrt{80}+9-\sqrt{80}+3t\sqrt[3]{{{\left( \sqrt{5}+2 \right)}^{2}}{{\left( \sqrt{5}-2 \right)}^{2}}} \\ =18+3t\sqrt[3]{{{\left[ {{\left( \sqrt{5} \right)}^{2}}-4 \right]}^{2}}} \\ =18+3t \\ \Leftrightarrow {{t}^{3}}-3t-18=0 \\ \Leftrightarrow \left( t-3 \right)\left( {{t}^{2}}+3t+6 \right)=0 \\ \Leftrightarrow \left[ \begin{align} & t=3 \\ & {{t}^{2}}+3t+6=0\,\left( VN \right) \\ \end{align} \right. \)
\(t = 3\) hay \(\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}=3\) (đpcm)