Simplify: 1+ the square root of 2 divided by 5 subtracted by  the square root of 2.

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To simplify this expression, we need to rationalize the denominator to get the square root completely into the numerator.  To rationalize, we multiply the numerator and denominator by the conjugate of the denominator.

`{1+sqrt2}/{5-sqrt2}`   multiply by conjugates

`={1+sqrt2}/{5-sqrt2} cdot {5+sqrt2}/{5+sqrt2}`   simplify numerator and denominator

`={5+5sqrt2+sqrt2+2}/{25-2}`   collect like terms

`={7+7sqrt2}/23`

The expression simplifies to `{7+7sqrt2}/23` .

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