5. Express $(A-B)^{-1}$ in terms of $x$, if $A=\frac{3 x^{-\frac{1}{3}}}{x^{\frac{2}{3}}-2 x^{-\frac{1}{3}}}$ and $B=\frac{x^{\frac{1}{3}}}{x^{\frac{4}{3}}-x^{\frac{1}{3}}}$. Simplify the expression.

## You have 120 minutes to solve.

Solve each problem independently on the provided paper, with each problem on a separate page. Write only your code on the sheets. Write your answers in pen, clearly and neatly. Draw graphs with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, strike it through.

Each solution will be graded on a scale of 0 to 6 points.

The National Competition Committee wishes you great success.

## 8th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 19, 2008

## Problems for 3rd Year Students
5. We simplify the expression $A$ by multiplying the fraction by $x^{\frac{1}{3}}$. We get $A=\frac{3}{x-2}$. The expression $B$ is multiplied by $x^{-\frac{1}{3}}$. We get $B=\frac{1}{x-1}$. We calculate the difference $A-B=\frac{3}{x-2}-\frac{1}{x-1}=\frac{2 x-1}{(x-2)(x-1)}$. We also write $(A-B)^{-1}=\frac{(x-2)(x-1)}{2 x-1}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-08.jpg?height=68&width=1636&top_left_y=2530&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-08.jpg?height=65&width=1639&top_left_y=2583&top_left_x=206)

The calculated difference $A-B=\frac{3}{x-2}-\frac{1}{x-1}=\frac{2 x-1}{(x-2)(x-1)} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots \ldots$ point

## Third Year