Question

The value of \( \frac{15}{\sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80}} \) is:

• A. \( \sqrt{5}(2 + \sqrt{2}) \)
• B. \( \sqrt{5}(5 + \sqrt{2}) \)
• C. \( \sqrt{5}(1 + \sqrt{2}) \)
• D. \( \sqrt{3}(3 + \sqrt{2}) \)

Hint:

When dealing with complex expressions involving square roots, rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Answer 1

The Correct Option is A

We are given the expression \( \frac{15}{\sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80}} \). We first simplify the terms in the denominator: \[ \sqrt{10} = \sqrt{2 \times 5}, \quad \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}, \quad \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}, \quad \sqrt{5} = \sqrt{5}, \quad \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \] Thus, the denominator becomes: \[ \sqrt{10} + 2\sqrt{5} + 2\sqrt{10} - \sqrt{5} - 4\sqrt{5} = 3\sqrt{10} - 3\sqrt{5} \] Now, factor out \( 3 \): \[ 3(\sqrt{10} - \sqrt{5}) \] Now the expression becomes: \[ \frac{15}{3(\sqrt{10} - \sqrt{5})} = \frac{5}{\sqrt{10} - \sqrt{5}} \] Multiply both the numerator and denominator by \( \sqrt{10} + \sqrt{5} \) to rationalize the denominator: \[ \frac{5(\sqrt{10} + \sqrt{5})}{(\sqrt{10} - \sqrt{5})(\sqrt{10} + \sqrt{5})} = \frac{5(\sqrt{10} + \sqrt{5})}{10 - 5} = \frac{5(\sqrt{10} + \sqrt{5})}{5} = \sqrt{10} + \sqrt{5} \] Thus, the final simplified result is \( \sqrt{5}(2 + \sqrt{2}) \).