Find the values the following correct to three places of decimals, it being given that `sqrt2 = 1.4142`, `sqrt3 = 1.732`, `sqrt5 = 2.2360`, `sqrt6 = 2.4495` and `sqrt10 = 3.162`
`(3 - sqrt5)/(3 + 2sqrt5)`
Solution
We know that rationalization factor for `3 + 2sqrt5` is `3 - sqrt5`. We will multiply numerator and denominator of the given expression `(3 - sqrt5)/(3 + 2sqrt5)` by `3 - 2sqrt5` to get
`(3 - sqrt5)/(3 + 2sqrt5) xx (3 - 2sqrt5)/(3 - 2sqrt5) = ((3)^2 - 3 xx 2 xx sqrt5 - 3 xx sqrt5 + 2 xx (sqrt5)^2)/((3)^2 - (2sqrt5)^2)`
` = (9 - 9sqrt5 + 10)/(9 - 20)`
`= (19 - 9sqrt5)/(-11)`
`= (9sqrt5 - 19)/11`
Putting the value of `sqrt5`, we get
`(9sqrt5 - 19)/11 = (9(2.236) - 19)/11`
`= (20.124 - 19)/11`
`= 1.124/11`
= 0.102
Hence the given expression is simplified to 0.102
