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If $\alpha, \beta$ are the roots of the equation $x^{2}+x+2=0$, then
what is $\frac{\alpha^{10}+\beta^{10}}{\alpha^{-10}+\beta^{-10}}$ equal to?
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what is $\frac{\alpha^{10}+\beta^{10}}{\alpha^{-10}+\beta^{-10}}$ equal to?
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The correct answer is:
1024
Here, $\alpha$ and $\beta$ are the roots of the equation $\alpha+\beta=-1$
$\alpha \beta=2$
$\frac{\alpha^{10}+\beta^{10}}{\alpha^{-10}+\beta^{-10}}=\frac{\alpha^{10}+\beta^{10}}{\frac{1}{\alpha^{10}}+\frac{1}{\beta^{10}}}$
$=\frac{\left(\alpha^{10}+\beta^{10}\right)(\alpha \beta)^{10}}{\left(\alpha^{10}+\beta^{10}\right)}=(\alpha \beta)^{10}$
$\therefore(\alpha \beta)^{10}=2^{10}=1024$
$\alpha \beta=2$
$\frac{\alpha^{10}+\beta^{10}}{\alpha^{-10}+\beta^{-10}}=\frac{\alpha^{10}+\beta^{10}}{\frac{1}{\alpha^{10}}+\frac{1}{\beta^{10}}}$
$=\frac{\left(\alpha^{10}+\beta^{10}\right)(\alpha \beta)^{10}}{\left(\alpha^{10}+\beta^{10}\right)}=(\alpha \beta)^{10}$
$\therefore(\alpha \beta)^{10}=2^{10}=1024$
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