A) 5.398
B) 4.258
C) 5.355
D) 3.855
Correct Answer: A
Solution :
(a): We have. \[\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}\] \[=\sqrt{10}+\sqrt{{{2}^{2}}\times 5}+\sqrt{{{2}^{2}}\times 10}-\sqrt{5}-\sqrt{{{2}^{4}}\times 5}\] \[=\sqrt{10}+2\sqrt{5}+2\sqrt{10}-\sqrt{5}-{{2}^{2}}-\sqrt{5}\] \[=\sqrt{10}+2\sqrt{10}+2\sqrt{5}-\sqrt{5}-4\sqrt{5}\] \[=(1+2)\sqrt{10}+(2-1-4)\sqrt{5}\] \[=3\sqrt{10}-3\sqrt{5}=3\left( \sqrt{10}-\sqrt{5} \right)\] \[\therefore \frac{15}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}\] \[=\frac{15}{3\left( \sqrt{10}-\sqrt{5} \right)}=\frac{5}{\sqrt{10}-\sqrt{5}}\] \[=\frac{8\left( \sqrt{10}-\sqrt{5} \right)}{\left( \sqrt{10}-\sqrt{5} \right)\left( \sqrt{10}+\sqrt{5} \right)}\] \[[\text{Multiplying}\,\text{and}\,\text{dividing}\,\text{by}\sqrt{10}+\sqrt{5}]\] \[=\frac{5\left( \sqrt{10}+\sqrt{5} \right)}{10-5}=\sqrt{10}+\sqrt{5}=3.162+2.236=5.398\]
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