2+sqrt{2}+frac{1}{2+sqrt{2}}+frac{1}{sqrt{2}- | vedclass

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Question

(A) Given expression: $2+\sqrt{2}+\frac{1}{2+\sqrt{2}}+\frac{1}{\sqrt{2}-2}$Combine the last two terms by finding a common denominator:$\frac{1}{2+\sq

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(A) Given expression: $2+\sqrt{2}+\frac{1}{2+\sqrt{2}}+\frac{1}{\sqrt{2}-2}$Combine the last two terms by finding a common denominator:$\frac{1}{2+\sqrt{2}}+\frac{1}{\sqrt{2}-2} = \frac{(\sqrt{2}-2) + (2+\sqrt{2})}{(2+\sqrt{2})(\sqrt{2}-2)}$Simplify the numerator and denominator:Numerator: $\sqrt{2}-2+2+\sqrt{2} = 2\sqrt{2}$Denominator: $(\sqrt{2})^2 - (2)^2 = 2 - 4 = -2$So,the sum of the last two terms is $\frac{2\sqrt{2}}{-2} = -\sqrt{2}$Now,substitute this back into the original expression:$2 + \sqrt{2} + (-\sqrt{2}) = 2$Therefore,the correct option is $A$.

$2+\sqrt{2}+\frac{1}{2+\sqrt{2}}+\frac{1}{\sqrt{2}-2}=?$

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