Question

Write the reciprocal of \[5 + \sqrt{2}\].

Answer 1

Given that,`5+sqrt2` it’s reciprocal is given as 

 `1/(5+sqrt2)`

It can be simplified by rationalizing the denominator. The rationalizing factor of `5+sqrt2` is ` 5 - sqrt2`, we will multiply numerator and denominator of the given expression `1/(5+sqrt2)`by, `5-sqrt2` to get

`1/(5+sqrt2) xx (5-sqrt2)/(5-sqrt2) = (5-sqrt2)/((5)^2  - (sqrt2)^2)`

`= (5-sqrt2) /( 25-2)` 

 ` =  (5- sqrt2 ) / 23 `

Hence reciprocal of the given expression is `(5- sqrt2 ) / 23 `.