If 8 α 9 = -72, -9 α 3 = 27 and -6 α 1 = 6 according to a certain rule, then what is the value of -3 α 6 ?

Introduction / Context:
This question introduces a custom operator α that combines two numbers in a consistent but nonstandard way. The examples show both positive and negative inputs, and the results suggest a simple relationship involving multiplication and a sign change. Our goal is to determine the rule that defines a α b and then apply it to the pair −3 and 6.

Given Data / Assumptions:

• 8 α 9 = -72.
• -9 α 3 = 27.
• -6 α 1 = 6.
• We must find the value of -3 α 6.
• The same operator definition applies to all three examples and to the unknown case.

Concept / Approach:
A natural starting point is to examine the product of the two numbers in each example. For 8 and 9, the product is 72, and the result is -72. For −9 and 3, the product is −27, and the result is 27, which is the negative of the product. Similarly, for −6 and 1, the product is −6 and the result is 6, again the negative of the product. This suggests that a α b is defined as the negative of the ordinary product a × b. Once this is confirmed, we can apply the same definition to −3 and 6.

Step-by-Step Solution:
Step 1: Compute the ordinary product for each example to test the pattern. Step 2: For 8 α 9, 8 × 9 equals 72. The given result is −72, which is the negative of 72. Step 3: For −9 α 3, −9 × 3 equals −27. The result is 27, which is the negative of −27. Step 4: For −6 α 1, −6 × 1 equals −6. The result is 6, again the negative of the product. Step 5: These three examples confirm that a α b = −(a × b). Step 6: Apply the rule to −3 α 6. First compute the ordinary product −3 × 6 = −18. Step 7: Now apply the negative sign required by the definition: −(−18) = 18. Step 8: Therefore, −3 α 6 equals 18.

Verification / Alternative check:
We can verify by checking that the rule produces each given result exactly and does so in a simple and consistent way. No other basic combination of addition and subtraction yields all three sample values without extra complications. The consistency of the negative product rule across all examples strongly supports its correctness and justifies using it for −3 and 6 as well.

Why Other Options Are Wrong:
The negative values −98 and −87 do not match the rule of taking the negative of the product, since the product of −3 and 6 is −18 and its negative is 18. The value 29 is unrelated to any simple combination of −3 and 6 that also fits the earlier examples. As a result, none of these alternatives can be correct if we keep the operator definition consistent.

Common Pitfalls:
Some candidates forget to apply the extra negative after computing the product and simply use a × b. Others miscalculate the product of negative numbers. Careful attention to signs when multiplying and then applying one more sign change is essential for full accuracy in such problems.

Final Answer:
Using the rule a α b = negative of a × b, the value of −3 α 6 is 18.