In this number analogy, 9 is related to 27 through an exponent pattern; using the same rule, select the number that correctly completes the analogy: 9 : 27 :: 64 : ?

Introduction / Context:
This problem is a number analogy question where each pair of numbers is linked by a power based rule. The first pair, 9 and 27, shows a particular exponent relationship. You must detect that relationship and then apply it to the second number, 64, to identify the correct matching term. Questions like this test your understanding of roots, powers and pattern recognition.

Given Data / Assumptions:

• The numbers involved are 9, 27, 64 and the unknown term.
• One pair is 9 : 27, the other is 64 : ?.
• The same mathematical operation must relate 9 to 27 and 64 to the missing number.
• We expect exponents or roots because 9 and 27 have simple power relationships.

Concept / Approach:
First, we express each number in the first pair in terms of prime powers. Nine is 3 squared, and twenty seven is 3 cubed. This indicates a change in exponent. A natural guess is that the second number is the first number raised to the power of three halves, since (3^2)^(3/2) equals 3^3. Once we verify that 9 raised to power three halves really equals 27, we can apply the same exponent to 64 and see which option matches.

Step-by-Step Solution:
Step 1: Write 9 as a power of 3. We have 9 = 3^2.Step 2: Write 27 as a power of 3. We have 27 = 3^3.Step 3: Observe that 3^3 can be obtained from 3^2 by raising 3^2 to power 3/2, because (3^2)^(3/2) = 3^(2 * 3/2) = 3^3.Step 4: Equivalently, 9^(3/2) = (square root of 9) cubed = 3^3 = 27. So the rule can be described as n raised to power 3/2.Step 5: Apply the same rule to 64. First find the square root of 64, which is 8.Step 6: Now cube the result: 8^3 = 512.Step 7: Therefore, under this exponent pattern, 64 maps to 512, which matches option C.

Verification / Alternative check:
We can also check other options using simpler patterns, such as multiplication by a constant. For instance, 9 multiplied by 3 gives 27, but 64 multiplied by 3 gives 192, which is not among the options. The exponent method works neatly for 9 and 27, and reproduces 512 from 64. Since 512 is 8 cubed and 8 is the square root of 64, the analogy is consistent and exact.

Why Other Options Are Wrong:
Option A, 225, is 15 squared and does not relate to 64 by the 3/2 exponent rule. Option B, 216, equals 6 cubed and would require a different operation. Option D, 324, is 18 squared and again does not match the discovered pattern. None of these preserve the relationship that 9 has with 27.

Common Pitfalls:
Many candidates only try linear operations like adding or multiplying by a fixed number and overlook exponent patterns. Others see that 27 is 3 cubed but fail to relate it to 9 as 3 squared. A good habit is to express the numbers with prime factors and compare exponents whenever powers seem likely. That quickly leads to the correct rule in this question.

Final Answer:
Following the same pattern, 64 is related to 512, so option C is correct.