Question

Simplify: \[\frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} - \frac{7 - 3\sqrt{5}}{3 - \sqrt{5}}\]

Answer 1

We know that rationalization factor for`3+sqrt5+`and  `3-sqrt5`are`3-sqrt5` and  `3+sqrt5`respectively. We will multiply numerator and denominator of the given expression  `(7+3sqrt5)/(3+sqrt5)`and  `(7-3sqrt5)/(3- sqrt5)` by` 3-sqrt5` and  `3+sqrt5`   respectively, to get

 `(7+3sqrt5)/(3+ sqrt5) xx  (3-sqrt5)/(3- sqrt5) - (7-3sqrt5)/(3- sqrt5) xx (3+sqrt5)/(3+ sqrt5) = (7xx3-7xxsqrt5+9xxsqrt5-3xx(sqrt5)^2)/ ((3)^2 - (sqrt5)^2) -(7xx3+7xxsqrt5-9xxsqrt5-3xx(sqrt5)^2)/ ((3)^2 - (sqrt5)^2) ` 

`=(21-7sqrt5+9sqrt5 - 3xx5)/(9-5) - (21+7sqrt5+9sqrt5 - 3xx5)/(9-5) `

`=(21+2sqrt5-15)/ 4 - (21-2sqrt5-15) /4`

`= (6+2sqrt5-6+2sqrt5)`

` = (4sqrt5 )/4`

` = sqrt5`