153 Let the function $f(x)=1-|1-2 x|, g(x)=x^{2}-2 x+1, x \in[0,1]$, and define
$$
F(x)=\left\{\begin{array}{ll}
f(x) & f(x) \geqslant g(x), \\
g(x) & f(x)<g(x) .
\end{array}\right.
$$

Then the number of real roots of the equation $F(x) \cdot 2^{x}=1$ is
153 3. As shown in Figure 13, the graphs of the functions $f(x)$ and $g(x)$ are plotted on the same coordinate plane, thus forming the graph of the function $F(x)$. Then, the graph of the function $y=\left(\frac{1}{2}\right)^{x}$ is plotted. It is easy to see that the graph of $y=\left(\frac{1}{2}\right)^{x}$ intersects the graph of $y=F(x)$ at 3 points. Therefore, the equation $F(x) \cdot 2^{x}=1$ has 3 real roots.