Question

For $x\ne 0,\pm 1,$ the expression $\frac{\frac{1}{{}_{x}2007}-\frac{1}{{}_{x}2009}}{\frac{1}{{}_{x}2008}-\frac{1}{{}_{x}2010}}$ is
equivalent to

• A. x
• B. x - 1
• C. $x^2-1$
• D. $\frac{1}{x}$

Answer 1

The correct option is A x

$\frac{\frac{1}{{}_{x}2007}-\frac{1}{{}_{x}2009}}{\frac{1}{{}_{x}2008}-\frac{1}{{}_{x}2010}}=\frac{x^{-2007}-x^{-2009}}{x^{-2008}-x^{-2010}}$
$=\frac{x^{-2009}(x^2-1)}{x^{-2010}(x^2-1)}$
$=\frac{x^{-2009}}{x^{-2010}}$
$=x^{-2209+2010}=x$
Therefore, option A is correct.