Algebraically simplify: -x^(-1) +1-(x-1)(x^-2) to ((1/x)-1)^2

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- Sep 16th 2010, 08:14 PMyessHow do you simplify this algebraically?
Algebraically simplify: -x^(-1) +1-(x-1)(x^-2) to ((1/x)-1)^2

- Sep 16th 2010, 08:36 PMpickslides
You have $\displaystyle \displaystyle \frac{-1}{x}+1-\frac{x-1}{x^2}$

I would suggest starting by making a common denominator of $\displaystyle x^2$ - Sep 17th 2010, 12:28 AMEducated
Remember that multiplying by negative powers is the same as dividing by that positive power.

So you would have the equation of:

$\displaystyle \dfrac{-1}{x}+1-\dfrac{x-1}{x^2}$

Make each of them have a common denominator by multiplying by corresponding values of x:

$\displaystyle = \dfrac{-x}{x^2}+\dfrac{x^2}{x^2}-\dfrac{x-1}{x^2}$

$\displaystyle =\dfrac{-x+x^2-x+1}{x^2}$

$\displaystyle =\dfrac{x^2-2x+1}{x^2}$

$\displaystyle =\dfrac{(1-x)^2}{x^2}$

Note, x^2-2x+1 can be factorised to (x-1)^2 or (1-x)^2. I used (1-x)^2 because it helps in getting the final answer you gave.

$\displaystyle =\left(\dfrac{1-x}{x}\right)^2$

$\displaystyle =(\frac{1}{x}-\frac{x}{x})^2$

$\displaystyle =(\frac{1}{x}-1)^2$ - Sep 17th 2010, 04:22 AMSoroban
Hello, yess!

Quote:

$\displaystyle \text{Simplify }\,-x^{-1} +1-(x-1)x^{-2}\:\text{ to }\:\left(\dfrac{1}{x}-1\right)^2$

We have: .$\displaystyle -\dfrac{1}{x} + 1 - \left(\dfrac{x-1}{x^2}\right) \;\;=\;\;-\dfrac{1}{x} + 1 - \left(\dfrac{x}{x^2} - \dfrac{1}{x^2}\right) $

. . . . . . $\displaystyle =\;\;-\dfrac{1}{x} + 1 - \dfrac{1}{x} + \dfrac{1}{x^2} \;\;=\;\;\dfrac{1}{x^2} - \dfrac{2}{x} + 1 $

Factor: . $\displaystyle \left(\dfrac{1}{x} - 1\right)^2$

- Sep 17th 2010, 04:46 AMArchie Meade
$\displaystyle -x^{-1}+1-(x-1)x^{-2}$

$\displaystyle =-\frac{1}{x}-(-1)-(x-1)x^{-2}$

$\displaystyle =-\left(\frac{1}{x}-1\right)+(1-x)x^{-2}$

$\displaystyle =-\left(\frac{1}{x}-1\right)+\frac{1}{x}(1-x)x^{-2}x$

$\displaystyle =-\left(\frac{1}{x}-1\right)+\left(\frac{1}{x}-1\right)x^{-1}$

$\displaystyle =\left(\frac{1}{x}-1\right)\left(x^{-1}-1\right)=\left(\frac{1}{x}-1\right)\left(\frac{1}{x}-1\right)$