If $2 * 3 = \sqrt{13}$ and $3 * 4 = 5$, then the value of $5 * 12$ is

### Concept & Logic Observe the relationship between the inputs and outputs to decode the hidden mathematical operation represented by the asterisk. $$ a * b = \sqrt{a^2 + b^2} $$ ### Step-by-Step Solution Let's test our hypothesis on the first given equation. $2 * 3 = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$. This matches the given condition perfectly. Now, verify the pattern with the second equation. $3 * 4 = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$. This also holds true. Finally, apply this decoded logic to the target expression. $5 * 12 = \sqrt{5^2 + 12^2}$ Calculate the squares inside the root: $= \sqrt{25 + 144}$ $= \sqrt{169}$ Evaluate the square root: $= 13$ ### Exam Strategy & Shortcut Recognize standard Pythagorean triplets immediately. The numbers $(3, 4, 5)$ and $(5, 12, 13)$ are classic sides of right-angled triangles. As soon as you see $3 * 4 = 5$, you should suspect the Pythagorean distance formula. Applying this instinctively gives $5, 12 \rightarrow 13$ without calculating the large squares manually. ### Common Pitfall Trying to force standard arithmetic operations (like direct addition or multiplication) on the numbers and getting stuck when they don't yield the given roots. Always consider squares and roots for aptitude operator puzzles. ### Final Answer Therefore, the correct answer is **13**.