In a certain operator puzzle, the operation ∆ is defined so that 1 ∆ (-5) = 5, 4 ∆ 7 = -28 and (-1) ∆ (-3) = -3. Based on this pattern, what is the value of 8 ∆ (-8)?

Introduction / Context:
This question belongs to the coding and operator puzzle category. Instead of ordinary arithmetic operations, a special binary operation ∆ is given through examples, and you must infer the hidden rule. Once the rule is discovered, you apply it to a new pair of numbers to find the required value. Such questions test pattern recognition and your ability to generalise from limited data.

Given Data / Assumptions:

• 1 ∆ (-5) = 5.
• 4 ∆ 7 = -28.
• (-1) ∆ (-3) = -3.
• We must find 8 ∆ (-8).
• The same single rule applies to all three given examples.

Concept / Approach:
The most direct approach is to guess a simple algebraic rule that fits every pair. A common pattern in such problems is that the result depends on the product of the two numbers, possibly with a sign or factor. We can assume a form like f(a, b) = k * a * b or f(a, b) = -a * b and test whether it fits all three given equalities. Once the rule is established, we apply it to the new input 8 and -8.

Step-by-Step Solution:
Step 1: Consider f(a, b) = -a * b.Step 2: For a = 1 and b = -5, f(1, -5) = -(1 * -5) = -(-5) = 5, which matches the first equation.Step 3: For a = 4 and b = 7, f(4, 7) = -(4 * 7) = -28, which matches the second equation.Step 4: For a = -1 and b = -3, f(-1, -3) = -((-1) * (-3)) = -(3) = -3, again matching the given result.Step 5: Therefore the operation is a ∆ b = -(a * b). For a = 8 and b = -8, we get 8 ∆ (-8) = -(8 * -8) = -(-64) = 64.

Verification / Alternative check:
We can briefly test other possible rules, such as a ∆ b = a * b or a ∆ b = a - b, but they fail for at least one of the three given examples. Since a ∆ b = -(a * b) correctly reproduces all three results, it is the unique simple rule needed here. Applying it to 8 and -8 produces the answer 64, so our reasoning is consistent.

Why Other Options Are Wrong:

• -64 would correspond to a ∆ b = a * b, which does not match the initial examples.
• -8 and 8 come from incorrect guesses such as a - b or b - a.
• 0 would require 8 * (-8) = 0 or some subtraction symmetry, which does not hold.

Common Pitfalls:
Many test takers try to mix addition and subtraction of the two numbers instead of focusing on the product. Another frequent error is to assume different rules for each example rather than a single consistent formula. Always verify your guessed rule against all given cases before applying it to the unknown case.

Final Answer:
The only rule that fits all examples is a ∆ b = -(a * b), so 8 ∆ (-8) = 64.