3. Simplify the expression

$$
(x-1)\left(\left(x^{-3}+x^{-2}+x^{-1}\right)\left(x^{-1}-1\right)-\left(x^{-2}-1\right)\left(x^{-2}+1\right)\right)^{-1}
$$
3. A power with a negative exponent in parentheses can be written as a fraction $(x-1)\left(\left(\frac{1}{x^{3}}+\frac{1}{x^{2}}+\frac{1}{x}\right)\left(\frac{1}{x}-1\right)-\left(\frac{1}{x^{2}}-1\right)\left(\frac{1}{x^{2}}+1\right)\right)^{-1}$. Expand the sums or differences to a common denominator $(x-1)\left(\left(\frac{1+x+x^{2}}{x^{3}}\right)\left(\frac{1-x}{x}\right)-\left(\frac{1-x^{2}}{x^{2}}\right)\left(\frac{1+x^{2}}{x^{2}}\right)\right)^{-1}$, perform the multiplication

$(x-1)\left(\frac{1+x+x^{2}-x-x^{2}-x^{3}}{x^{4}}-\frac{1-x^{4}}{x^{4}}\right)^{-1}$. Subtract the fractions in the parentheses $(x-1)\left(\frac{x^{4}-x^{3}}{x^{4}}\right)^{-1}$. Write the reciprocal $(x-1)\left(\frac{x^{4}}{x^{3}(x-1)}\right)$, simplify and get the result $x$.

Writing powers as fractions $\left(\frac{1}{x^{3}}+\frac{1}{x^{2}}+\frac{1}{x}\right)\left(\frac{1}{x}-1\right)-\left(\frac{1}{x^{2}}-1\right)\left(\frac{1}{x^{2}}+1\right) \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ point

Expanding fractions to a common denominator $\left(\frac{1+x+x^{2}}{x^{3}}\right)\left(\frac{1-x}{x}\right)-\left(\frac{1-x^{2}}{x^{2}}\right)\left(\frac{1+x^{2}}{x^{2}}\right) \ldots \ldots \ldots \ldots .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-07.jpg?height=66&width=1639&top_left_y=704&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-07.jpg?height=77&width=1639&top_left_y=761&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-07.jpg?height=75&width=1636&top_left_y=819&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-07.jpg?height=55&width=1636&top_left_y=872&top_left_x=207)