Question

How do you rationalize the denominator and simplify `7/(1+sqrt2)`?

Answer 1

`= -7 (1 - sqrt2)`

Explanation:

`7 / ( 1 + sqrt2)`

We rationalize , by multiplying the expression by conjugate of the denominator , `(1 + sqrt2) = color(blue)( 1 - sqrt2`

`= (7 * color(blue)( (1 - sqrt2))) / (( 1 + sqrt2) * color(blue)(( 1 - sqrt2))`

`= (7 * color(blue)( (1 )) + 7 * color(blue)((- sqrt2))) / (( 1 + sqrt2) * color(blue)(( 1 - sqrt2))`

• Applying property:
  `color(blue)((a + b ) (a-b) = a^2 - b^2` to the denominator.

`= ((7 - 7 sqrt2)) / (( 1 ^2 - (sqrt2)^2)`

`= ((7 - 7 sqrt2)) / (( 1 - 2 )`

`= (7 (1 - sqrt2)) / (-1 )`

`= -7 (1 - sqrt2)`