If \(a=\sqrt{11}+\sqrt{5}\) and \(b=\sqrt{11}-\sqrt{5}\), what is the value of \(ab\)?
Answer and explanation
Correct answer: 6
These are conjugate terms. \(ab=(\sqrt{11}+\sqrt{5})(\sqrt{11}-\sqrt{5})=(\sqrt{11})^2-(\sqrt{5})^2=11-5=6\). A common wrong choice is \(\sqrt{55}\), which equals \(\sqrt{11}\cdot\sqrt{5}\) but is not the product of the conjugates. Option 16 arises from adding 11 and 5 (not correct for multiplication), and \(2\sqrt{55}\) is an incorrect doubled product. Exam tip: recognize conjugates and apply the difference-of-squares identity \(a^2-b^2\) to remove radicals quickly.
Frequently asked questions
What is the correct answer to this question?
6
Why is this the correct answer?
These are conjugate terms. \(ab=(\sqrt{11}+\sqrt{5})(\sqrt{11}-\sqrt{5})=(\sqrt{11})^2-(\sqrt{5})^2=11-5=6\). A common wrong choice is \(\sqrt{55}\), which equals \(\sqrt{11}\cdot\sqrt{5}\) but is not the product of the conjugates. Option 16 arises from adding 11 and 5 (not correct for multiplication), and \(2\sqrt{55}\) is an incorrect doubled product. Exam tip: recognize conjugates and apply the difference-of-squares identity \(a^2-b^2\) to remove radicals quickly.
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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