Let's find the roots of the following equation with a precision of two decimal places by systematic trial and error:

$$
(\sqrt[3]{3})^{x}=x
$$
In a graphical solution, the approximate value of the root is given by the abscissa of the point where the graphs of the functions on both sides of (1) intersect. The value on the left side for $x=0,1,2,3,4$ is respectively $1,1,44,2,08$, $3,4,33$ (Figure 1); the graph has a common point with the line $y=x$ at the abscissa $x=3$, which is the root $x_{1}=3$ of (1). The "curvature" of the graph suggests that there might be another point where it intersects the line $y=x$.
![](https://cdn.mathpix.com/cropped/2024_05_02_650b61a80f504031c6acg-1.jpg?height=546&width=1154&top_left_y=364&top_left_x=478)

Figure 1 and Figure 2

On the other hand, the increasingly steeper rise of the exponential curve suggests that there are no more than two common points.

The approximate value of $x_{2}$ is easier to find based on the following transformation of (1):

$$
\begin{gathered}
x \cdot \frac{\lg 3}{3}=\lg x \\
\frac{\lg x}{x}=\frac{\lg 3}{3} \approx \frac{0,4771}{3} \approx 0,1590
\end{gathered}
$$

(rounded), and the graph of the right side is also a straight line. Denoting the left side of (2) by $f(x)$ and the right side by $c$,

$$
f(2)=f(4)=0,1505c
$$

from this, $2c, \quad f(2,4) \approx 0,1584c
\end{gathered}
$$

thus, with the required precision, $x_{2}=2,48$.

Judit Tegze (Budapest, Kölcsey F. Gymnasium, III. grade)

Ádám Szentgáli (Budapest, Ady E. Gymnasium, IV. grade)