If $\oplus$ is an operation such that $a \oplus b = \begin{cases} 2a, & \text{when } a > b \\ a + b, & \text{when } a < b \\ a^2, & \text{when } a = b \end{cases}$ then $\left[ \frac{(5 \oplus 7) + (4 \oplus 4)}{3(5 \oplus 5) - (15 \oplus 11) - 3} \right]$ is equal to

### Concept & Logic This involves evaluating a piecewise mathematical function. For each operation in parentheses, you must first compare the two numbers to determine which of the three conditional rules applies, and then execute that specific rule. ### Step-by-Step Solution * **Given Rules:** * If $a > b$, use $2a$ * If $a < b$, use $a + b$ * If $a = b$, use $a^2$ * **Step 1: Evaluate Numerator Terms** * $(5 \oplus 7)$: Here, $5 < 7$ ($a < b$). So use $a + b = 5 + 7 = 12$. * $(4 \oplus 4)$: Here, $4 = 4$ ($a = b$). So use $a^2 = 4^2 = 16$. * Numerator Total $= 12 + 16 = 28$. * **Step 2: Evaluate Denominator Terms** * $(5 \oplus 5)$: Here, $5 = 5$ ($a = b$). So use $a^2 = 5^2 = 25$. * $(15 \oplus 11)$: Here, $15 > 11$ ($a > b$). So use $2a = 2(15) = 30$. * Denominator Total $= 3(25) - 30 - 3 = 75 - 30 - 3 = 42$. * **Step 3: Calculate Final Fraction** $$\text{Result} = \frac{28}{42}$$ Divide numerator and denominator by their greatest common divisor, 14: $$\text{Result} = \frac{2}{3}$$ ### Exam Strategy & Shortcut Process the conditionals systematically to avoid cognitive overload. Write down the intermediate values directly above the expression on your scratch paper: $[ (12) + (16) ] / [ 3(25) - (30) - 3 ]$ Doing this explicitly prevents you from accidentally combining the multiplier $3$ into the conditional logic of $(5 \oplus 5)$. ### Common Pitfall Misreading the greater-than/less-than signs or applying the rule to the wrong variable. For example, in $(15 \oplus 11)$ where $a > b$, the rule is $2a$. Some students mistakenly calculate $2b = 22$. Always trace back to the exact definition carefully. ### Final Answer **Therefore, the correct answer is $\frac{2}{3}$.**