1. The problem is to simplify the expression $\lceil x \rceil \lceil x \rceil$. 2. The ceiling function $\lceil x \rceil$ returns the smallest integer greater than or equal to $x$. 3. Since the expression is the product of $\lceil x \rceil$ with itself, it can be written as: $$\lceil x \rceil \times \lceil x \rceil = (\lceil x \rceil)^2$$ 4. This means the expression is simply the square of the ceiling of $x$. 5. Therefore, the simplified form is: $$(\lceil x \rceil)^2$$ This is the final answer, representing the square of the smallest integer greater than or equal to $x$.