Step-by-step explanation:
To determine whether a number is a perfect cube, we need to find if there is an integer \( n \) such that \( n^3 = \text{number} \).
Let's analyze each of the given numbers:
### 1. Checking if 12167 is a perfect cube:
Calculate the cube root of 12167:
\[
\sqrt[3]{12167} = 23
\]
Since \( 23^3 = 23 \times 23 \times 23 = 12167 \), 12167 is a perfect cube.
### 2. Checking if 11109 is a perfect cube:
Calculate the cube root of 11109:
\[
\sqrt[3]{11109} \approx 22.28
\]
Since the cube root is not an integer, 11109 is not a perfect cube.
### 3. Checking if 85184 is a perfect cube:
Calculate the cube root of 85184:
\[
\sqrt[3]{85184} \approx 44.25
\]
Since the cube root is not an integer, 85184 is not a perfect cube.
### Conclusion
Among the numbers given:
- 12167 is a perfect cube.
- 11109 is not a perfect cube.
- 85184 is not a perfect cube.
So, the perfect cube among the given numbers is 12167.
