Question

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Answer 1

N=\(1+\sqrt{\left(1-\sqrt{2}\right)^2}+\dfrac{1}{2}\sqrt{8}\)

=\(1+\left|1-\sqrt{2}\right|+\dfrac{1}{2}\cdot2\sqrt{2}\)

=\(1+\sqrt{2}-1+\sqrt{2}\)=\(2\sqrt{2}\)

M=\(\dfrac{1}{\sqrt{2}+1}+\dfrac{1}{\sqrt{2}-1}\)

=\(\dfrac{\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}+\dfrac{\left(\sqrt{2}+1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}\)

=\(\sqrt{2}-1+\sqrt{2}+1=2\sqrt{2}\)

N+M=\(2\sqrt{2}+2\sqrt{2}=4\sqrt{2}\)