If \(a=2-\sqrt{5}\) and \(b=2+\sqrt{5}\), what type of number is \(a+b\)?
Answer and explanation
Correct answer: Rational number
Compute the sum: \(a+b=(2-\sqrt{5})+(2+\sqrt{5})=2+2=4\). Since \(4\) is an integer, it is rational. Option B (irrational) is incorrect because the irrational parts \(-\sqrt{5}\) and \(+\sqrt{5}\) cancel out. Option C (non-real) is wrong because both terms are real numbers so their sum is real. Option D (always negative) is wrong because the sum is +4, not negative. Exam tip: Look for conjugate pairs — the radical parts often cancel when you add conjugates, leaving a rational result.
Frequently asked questions
What is the correct answer to this question?
Rational number
Why is this the correct answer?
Compute the sum: \(a+b=(2-\sqrt{5})+(2+\sqrt{5})=2+2=4\). Since \(4\) is an integer, it is rational. Option B (irrational) is incorrect because the irrational parts \(-\sqrt{5}\) and \(+\sqrt{5}\) cancel out. Option C (non-real) is wrong because both terms are real numbers so their sum is real. Option D (always negative) is wrong because the sum is +4, not negative. Exam tip: Look for conjugate pairs — the radical parts often cancel when you add conjugates, leaving a rational result.
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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