5. Find the integer solution of the equation

$$
\sqrt{3^{x} \sqrt[3]{9^{x} \sqrt[x]{27^{-1}}}}=9 \cdot \sqrt[3]{3}
$$

## You have 120 minutes to solve the problems.

Solve each problem independently on the provided paper, with each problem on a separate page. Write only your code on the sheets. Use a pen, write 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.

## 9th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools National Competition, April 18, 2009

## Problems for 4th Year Students
5. We write the equation in the form of a product of powers, multiply the powers with the same bases $3^{\frac{x}{2}+\frac{x}{3}-\frac{1}{2 x}}=$ $3^{2+\frac{1}{3}}$. We equate the exponents and get the equation $5 x^{2}-14 x-3=0$. The solutions to the equation are $x_{1}=3$ and $x_{2}=-\frac{1}{5}$. We choose the integer solution $x=3$.

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-12.jpg?height=66&width=1639&top_left_y=1138&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-12.jpg?height=74&width=1642&top_left_y=1191&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-12.jpg?height=51&width=1642&top_left_y=1251&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-12.jpg?height=52&width=1642&top_left_y=1299&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-12.jpg?height=68&width=1642&top_left_y=1345&top_left_x=207)

## Fourth Year