√(a² + b²) is not a + b: the radical shortcut that fails
√(9 + 16) = 3 + 4 = 7 is beautiful, symmetric, and wrong. It's 5 — and the SAT knows 7 is tempting.
Why it happens
Radicals DO distribute over multiplication (√(ab) = √a·√b), and the pattern generalizes in your head to addition, where it's false.
See the trap
Evaluate √(9 + 16).
√9 + √16 = 3 + 4 = 7.
√25 = 5.
How to actually fix it
- 1Add first, root last — always simplify inside the radical before splitting anything.
- 2Splitting is only legal across × and ÷, never + and −.
- 33-4-5 triangles are the instant counterexample to recall.
FAQ
Same for (a + b)²?
Mirror image of the same illusion: (a + b)² = a² + 2ab + b², not a² + b². The missing 2ab term is its own distractor family.
How do I stop repeating this mistake?
Reading the explanation isn't enough — your brain needs to beat the trap on a fresh question. Practice the skill in the kaoshen.co question bank: miss one and it is logged with the exact error, then brought back to you three days later and again after fourteen.
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