9: 512::8:?

Options

A. 512
B. 415
C. 343
D. 216

Explanation

To solve the analogy problem \(9 : 512 :: 8 : ?\), we need to identify the relationship between the numbers in the first pair and then apply that same relationship to the second pair.

### Step-by-Step Solution:

1. **Identify the relationship in the first pair (9 : 512)**:
- We observe that \(512\) can be expressed as \(9^3\) (since \(9\) is \(3^2\) and \(3^3 = 27\), which is not directly applicable here). Instead, we can think of \(512\) as \(8^3\).
- However, we notice that \(512 = 8^3\) and \(9 = 8 + 1\).
- So, we can express the relationship as:
\[ 9 = 8 + 1 \quad \text{and} \quad 512 = 8^3 \]

2. **Apply the same relationship to the second pair (8 : ?)**:
- We take \(8\) and apply the same logic. We need to find \(? = 7^3\) (since \(8 - 1 = 7\)).
- Calculate \(7^3\):
\[ 7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343 \]

3. **Conclusion**:
- Therefore, the value of \(?\) is \(343\).
- The final answer is:
\[ 9 : 512 :: 8 : 343 \]

### Final Answer:
\[ ? = 343 \]