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\frac{1}{\frac{2}{2}+\frac{\sqrt{2}}{2}}
To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.
\frac{1}{\frac{2+\sqrt{2}}{2}}
Since \frac{2}{2} and \frac{\sqrt{2}}{2} have the same denominator, add them by adding their numerators.
\frac{2}{2+\sqrt{2}}
Divide 1 by \frac{2+\sqrt{2}}{2} by multiplying 1 by the reciprocal of \frac{2+\sqrt{2}}{2}.
\frac{2\left(2-\sqrt{2}\right)}{\left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right)}
Rationalize the denominator of \frac{2}{2+\sqrt{2}} by multiplying numerator and denominator by 2-\sqrt{2}.
\frac{2\left(2-\sqrt{2}\right)}{2^{2}-\left(\sqrt{2}\right)^{2}}
Consider \left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{2\left(2-\sqrt{2}\right)}{4-2}
Square 2. Square \sqrt{2}.
\frac{2\left(2-\sqrt{2}\right)}{2}
Subtract 2 from 4 to get 2.
2-\sqrt{2}
Cancel out 2 and 2.