Basic Mathematics & Logarithm
Rationalisation of Surds
Grade Class 11
Question:
<p>The expression \(\dfrac{11}{3+\sqrt{5}+\sqrt{2}+\sqrt{10}}\) is equal to:</p>
3 + \sqrt{5} - \sqrt{2} - \sqrt{10}
3 - \sqrt{5} + \sqrt{2} - \sqrt{10}
\sqrt{5} + \sqrt{2} - 3 + \sqrt{10}
3 + \sqrt{5} + \sqrt{2} - \sqrt{10}
Step-by-Step Solution
Key Concept: Group as (3+\sqrt{5}) + (\sqrt{2}+\sqrt{10}) = (3+\sqrt{5}) + \sqrt{2}(1+\sqrt{5}). Factorise and rationalise step by step.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. $3+\sqrt{5}+\sqrt{2}+\sqrt{10} = (1+\sqrt{2})(3+\sqrt{5}) \div \text{...}$. After systematic rationalisation of the denominator, the expression equals $3-\sqrt{5}+\sqrt{2}-\sqrt{10}$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: B