5. Simplify the expression

$$
\frac{x^{0.5}+1}{x+x^{0.5}+1} : \frac{1}{x^{1.5}-1}
$$

Solve each problem independently on the attached paper, and each problem on its own page.

Do not sign the sheets, write only your code.

Write your solution with a pen clearly and neatly. Each problem's solution will be graded from 0 to 6 points.

You have 120 minutes to solve the problems.

4th National Mathematics Competition for Students of Secondary Technical and Vocational Schools, Maribor, April 17, 2004

## PROBLEMS FOR 3RD YEAR STUDENTS
5. Potences with rational exponents can be written with roots. Thus, $\frac{\sqrt{x}+1}{x+\sqrt{x}+1}: \frac{1}{\sqrt{x^{3}}-1}=$ $\frac{\sqrt{x}+1}{x+\sqrt{x}+1} \cdot \frac{\sqrt{x^{3}}-1}{1}$. We multiply the fractions and get $\frac{\sqrt{x^{4}}+\sqrt{x^{3}}-\sqrt{x}-1}{x+\sqrt{x}+1}$. We simplify the numerator to $\frac{x^{2}+x \sqrt{x}-\sqrt{x}-1}{x+\sqrt{x}+1}$, then we transform it to $\frac{(x-1)(x+1)+\sqrt{x}(x-1)}{x+\sqrt{x}+1}$ and factor out the common factor in the numerator: $\frac{(x-1)(x+1+\sqrt{x})}{x+\sqrt{x}+1}$. After simplifying, we get $x-1$.

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-10.jpg?height=126&width=1642&top_left_y=220&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-10.jpg?height=63&width=1642&top_left_y=320&top_left_x=207)

Simplification of the numerator: $x^{2}+x \sqrt{x}-\sqrt{x}-1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Factoring out the common factor in the numerator: $(x-1)(x+1)+\sqrt{x}(x-1) \ldots \ldots \ldots \ldots . .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-10.jpg?height=112&width=1642&top_left_y=481&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-10.jpg?height=54&width=1636&top_left_y=573&top_left_x=207)

## Third Year