Is it possible that a set consisting of $23$ real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly $2422$?
1. **Define the sets \( A \), \( B \), and \( C \):**
   - Let \( A = \{2^{\frac{1}{2885}}, 2^{\frac{2^1}{2885}}, 2^{\frac{2^2}{2885}}, \dots, 2^{\frac{2^{10}}{2885}}\} \).
   - Let \( B = \{2 \cdot 2^{\frac{1}{2885}}, 2 \cdot 2^{\frac{2^1}{2885}}, 2 \cdot 2^{\frac{2^2}{2885}}, \dots, 2 \cdot 2^{\frac{2^{10}}{2885}}\} \).
   - Let \( C = \{3 \cdot 2^{\frac{2}{2885}}\} \).

2. **Verify the number of elements in each set:**
   - \( A \) and \( B \) each contain \( 11 \) elements.
   - \( C \) contains \( 1 \) element.
   - Therefore, the total number of elements in \( A \cup B \cup C \) is \( 11 + 11 + 1 = 23 \).

3. **Determine the condition for the product of elements in a subset to be rational:**
   - The product of elements in a subset is rational if and only if the sum of the exponents of \( 2 \) in the product is an integer.
   - For elements in \( A \), the exponents are \( \frac{1}{2885}, \frac{2^1}{2885}, \frac{2^2}{2885}, \dots, \frac{2^{10}}{2885} \).
   - For elements in \( B \), the exponents are \( 1 + \frac{1}{2885}, 1 + \frac{2^1}{2885}, 1 + \frac{2^2}{2885}, \dots, 1 + \frac{2^{10}}{2885} \).
   - For the element in \( C \), the exponent is \( \frac{2}{2885} \).

4. **Count the number of nonempty subsets whose product is rational:**
   - We need to count the number of solutions to \( a + b + c = 2885 \) where \( a, b \in \{0, 1, \dots, 2047\} \) and \( c \in \{0, 2\} \).
   - When \( c = 0 \), the number of solutions is \( 2047 - 838 + 1 = 1210 \).
   - When \( c = 2 \), the number of solutions is \( 2047 - 836 + 1 = 1212 \).
   - Therefore, the total number of solutions is \( 1210 + 1212 = 2422 \).

5. **Conclusion:**
   - The set \( A \cup B \cup C \) satisfies the condition that the number of nonempty subsets whose product of the elements is a rational number is exactly \( 2422 \).

The final answer is \( \boxed{2422} \)