Question

How do you rationalize the denominator and simplify `1/ (1+sqrt2)`?

Answer 1

`-1+sqrt2`

Explanation:

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`"numerator/denominator by the conjugate of the"`
`"denominator"`

`"the conjugate of "1+sqrt2" is " 1 color(red)(-)sqrt2`

`=(1-sqrt2)/((1+sqrt2)(1-sqrt2))`

`=(1-sqrt2)/(1^2-(sqrt2)^2)=(1-sqrt2)/(-1)=-1+sqrt2`

Answer 2

`\sqrt2-1`

Explanation:

Given that

`1/{1+\sqrt2}`

`=1/{\sqrt2+1}`

`={\sqrt2-1}/{(\sqrt2+1)(\sqrt2-1)}`

`={\sqrt2-1}/{(\sqrt2)^2-1^2}`

`={\sqrt2-1}/{2-1}`

`=\sqrt2-1`