Question 12

What is the value of $$(2+\sqrt{2})+\left(\frac{1}{2+\sqrt{2}}\right)+\left(\frac{1}{2-\sqrt{2}}\right)+(2-\sqrt{2})$$?

Solution

$$(2+\sqrt{2})+\left(\frac{1}{2+\sqrt{2}}\right)+\left(\frac{1}{2-\sqrt{2}}\right)+(2-\sqrt{2})$$

$$=(2+\sqrt{2})+\left(\frac{2-\sqrt{2}}{4-2}\right)+\left(\frac{2+\sqrt{2}}{4-2}\right)+(2-\sqrt{2})$$

$$=\left(2+2\right)+\left(\frac{2-\sqrt{2}+2+\sqrt{2}}{2}\right)\ .$$

$$=4+\left(\frac{4}{2}\right)\ .$$

$$=6\ .$$

D is correct choice.


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