2. How many roots does the equation

$\sqrt[3]{|x|}+10[x]=10 x$ ? ( $[x]$ - the integer part of the number $x$, i.e., $[x] \in Z,[x] \leq x<[x]+1$).

(5 points)
# Solution:

$\frac{\sqrt[3]{|x|}}{10}=\{x\},\{x\}=x-[x]$ Since $\{x\} \in[0 ; 1)$, then $x \in(-1000 ; 1000)$.

![](https://cdn.mathpix.com/cropped/2024_05_06_555865d131a60f67fb06g-3.jpg?height=914&width=1422&top_left_y=180&top_left_x=134)

On the interval $[0 ; 1)$, the equation has 2 roots

$\sqrt[3]{x}=10 x, x=1000 x^{3}, x=0, x=\sqrt{0.001}$. On all other intervals $[n, n+1)$, where $n=-1000, \ldots,-1,1, \ldots, 999$, the equation has one root. In total, we have 2000 roots.

Answer: 2000