In a certain coded pattern, expressions of the form a + b x c + d are not evaluated normally. Instead, 19 + 2 x 2 + 3 is equal to 369 and 23 + 2 x 6 + 2 is equal to 565. Using the same hidden rule, what is the value of 7 + 3 x 11 + 3 ?

Introduction / Context:
This question presents equations where normal arithmetic does not apply. Instead of evaluating 19 + 2 x 2 + 3 in the usual way, the result is given as 369. Similarly, 23 + 2 x 6 + 2 is given as 565. This clearly indicates that the symbols are being used as part of a coding pattern. The goal is to identify the hidden relationship between the four numbers in each expression and the three digit result, and then use this relationship to find the value corresponding to 7 + 3 x 11 + 3. This tests deeper pattern recognition and comfort with powers and squares.

Given Data / Assumptions:

• 19 + 2 x 2 + 3 = 369 (coded form).
• 23 + 2 x 6 + 2 = 565 (coded form).
• The same rule applies for all such expressions.
• The expression to decode is 7 + 3 x 11 + 3.
• The plus and multiplication signs are part of the pattern rather than ordinary operations.

Concept / Approach:
A good starting point is to look at the first number and see whether its square is related to the coded result. For the first expression, 19^2 = 361, which is very close to 369. The difference is 369 - 361 = 8. Now observe that 2^3 = 8, and 2 and 3 are the second and fourth numbers in the expression. For the second expression, 23^2 = 529. The difference 565 - 529 = 36, and 6^2 = 36 using the third number 6 and the fourth number 2 as the exponent (6^2). These observations suggest that the coded result is of the form (first number)^2 + (third number)^(fourth number). Once this rule is confirmed for both given equations, we can apply it to 7, 3, 11 and 3.

Step-by-Step Solution:
Step 1: For 19 + 2 x 2 + 3, treat the numbers as a = 19, b = 2, c = 2, d = 3. Step 2: Compute a^2 = 19^2 = 361. Step 3: Compute c^d = 2^3 = 8. Step 4: Add these: 361 + 8 = 369, matching the coded result. Step 5: For 23 + 2 x 6 + 2, a = 23, c = 6, d = 2. Compute a^2 = 23^2 = 529. Step 6: Compute c^d = 6^2 = 36. Step 7: Sum: 529 + 36 = 565, again matching the coded result exactly. Step 8: Now apply the rule to 7 + 3 x 11 + 3, which gives a = 7, c = 11, d = 3. Step 9: Compute a^2 = 7^2 = 49. Step 10: Compute c^d = 11^3 = 1331. Step 11: Add these: 49 + 1331 = 1380.

Verification / Alternative check:
The rule coded result = a^2 + c^d fits both original examples without any adjustment, which makes it a very strong candidate for the intended pattern. Alternative constructions, such as using the second number as the base or adding all four numbers together, do not reproduce both 369 and 565. Rechecking the cube of 11 (11 * 11 * 11 = 1331) and adding 49 confirms that 1380 is accurate. Because the pattern is consistent and exact, 1380 is the correct coded value for the third expression.

Why Other Options Are Wrong:
Options 1674, 1268, 1496 and 1532 do not arise from a^2 + c^d for the numbers 7 and 11. They would require changing the pattern in a way that breaks the earlier examples, for instance by using different exponents or summing all four numbers. These distractions are meant to catch those who attempt ad hoc calculations or partial observations rather than establishing a single rule that works in every given case.

Common Pitfalls:
One common mistake is to treat the plus and multiplication signs as normal and compute 19 + 2 x 2 + 3 directly, which yields 26 instead of 369. Another error is to suspect a pattern involving sums of digits or concatenation without checking how the first number's square naturally appears in the result. Because the coded outputs are relatively large, recognising squares and powers is crucial. Always verify that your proposed rule explains every given example before applying it to the target expression.

Final Answer:
Using the pattern coded value = (first number)^2 + (third number)^(fourth number), the value of 7 + 3 x 11 + 3 is 1380.