Question:
If the operation $\oplus$ is defined by $a \oplus b = a^2 + b^2$ for all real numbers $a$ and $b$, then $(2 \oplus 3) \oplus 4 =$
If the operation $\oplus$ is defined by $a \oplus b = a^2 + b^2$ for all real numbers $a$ and $b$, then $(2 \oplus 3) \oplus 4 =$
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Always treat custom binary operations exactly like a function $f(a, b)$. Evaluate the innermost "function call" first before moving outward.
Updated On: May 6, 2026
- \( 120 \)
- \( 185 \)
- \( 175 \)
- \( 129 \)
- \( 312 \)
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The Correct Option is B
Solution and Explanation
Concept:
Binary operations are rules that combine two elements to produce a third. To solve nested operations, follow the order of operations by resolving the expression inside the parentheses first, then using that result for the outer operation.
Step 1: Evaluate the expression inside the parentheses.
Using the definition $a \oplus b = a^2 + b^2$: \[ 2 \oplus 3 = 2^2 + 3^2 = 4 + 9 = 13 \]
Step 2: Perform the outer operation.
Substitute the result from Step 1 back into the full expression: \[ (2 \oplus 3) \oplus 4 = 13 \oplus 4 \] Apply the definition again: \[ 13 \oplus 4 = 13^2 + 4^2 = 169 + 16 = 185 \]
Step 1: Evaluate the expression inside the parentheses.
Using the definition $a \oplus b = a^2 + b^2$: \[ 2 \oplus 3 = 2^2 + 3^2 = 4 + 9 = 13 \]
Step 2: Perform the outer operation.
Substitute the result from Step 1 back into the full expression: \[ (2 \oplus 3) \oplus 4 = 13 \oplus 4 \] Apply the definition again: \[ 13 \oplus 4 = 13^2 + 4^2 = 169 + 16 = 185 \]
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