Which option is the correct expansion of \( (\sqrt{6}+\sqrt{2})^2 \)?
Answer and explanation
Correct answer: 8+4\sqrt{3}
\( (\sqrt{6}+\sqrt{2})^2 = (\sqrt{6})^2 + 2\cdot\sqrt{6}\cdot\sqrt{2} + (\sqrt{2})^2 = 6 + 2\sqrt{12} + 2.\) Since \(\sqrt{12}=2\sqrt{3}\), the middle term becomes \(2\sqrt{12}=4\sqrt{3}\). Thus the expansion is \(8+4\sqrt{3}\). Option B (\(8+2\sqrt{3}\)) reflects the common mistake of treating \(\sqrt{12}\) as \(\sqrt{3}\), so it is incorrect. Exam tip: apply \((a+b)^2=a^2+2ab+b^2\) and simplify radicals by extracting perfect-square factors.
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What is the correct answer to this question?
8+4\sqrt{3}
Why is this the correct answer?
\( (\sqrt{6}+\sqrt{2})^2 = (\sqrt{6})^2 + 2\cdot\sqrt{6}\cdot\sqrt{2} + (\sqrt{2})^2 = 6 + 2\sqrt{12} + 2.\) Since \(\sqrt{12}=2\sqrt{3}\), the middle term becomes \(2\sqrt{12}=4\sqrt{3}\). Thus the expansion is \(8+4\sqrt{3}\). Option B (\(8+2\sqrt{3}\)) reflects the common mistake of treating \(\sqrt{12}\) as \(\sqrt{3}\), so it is incorrect. Exam tip: apply \((a+b)^2=a^2+2ab+b^2\) and simplify radicals by extracting perfect-square factors.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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