Analogy Pattern Identification

The question presents an analogy: \(145 : 15^{4} :: 532 : ?\). We need to find the relationship between the first pair (145 and \(15^{4}\)) and apply it to the third term (532) to find the fourth term.

Deriving the Relationship

Let's examine the first pair: 145 and \(15^{4}\).

• The number 145 consists of digits 1, 4, and 5.
• The base number in the second term is 15. This can be formed by taking the first digit (1) and the last digit (5) of 145, forming the number 15 (i.e., \(1 \times 10 + 5\)).
• The exponent is 4. This matches the middle digit of 145.
• Therefore, the rule appears to be: (First Digit × 10 + Last Digit)^(Middle Digit).
• Applying this to 145: \( (1 \times 10 + 5)^{4} = 15^{4} \). This matches the given analogy.

Applying the Pattern to 532

Now, let's apply the same rule to the number 532.

• The digits are 5, 3, and 2.
• First Digit = 5
• Middle Digit = 3
• Last Digit = 2
• Using the rule: (First Digit × 10 + Last Digit)^(Middle Digit)
• Calculate the base: \( 5 \times 10 + 2 = 52 \).
• The exponent is the middle digit: 3.
• So, the result is \( 52^{3} \).

Conclusion

The missing term in the analogy is \(52^{3}\), which corresponds to Option B.