Question

Express the following with rational denominator:

`(6 - 4sqrt2)/(6 + 4sqrt2)`

Answer 1

We know that rationalization factor for `6 + 4sqrt2` is `6 - 4sqrt2`. We will multiply numerator and denominator of the given expression `(6 - 4sqrt2)/(6 + 4sqrt2)` by `6 - 4sqrt2` to get

`(6 - 4sqrt2)/(6 + 4sqrt2) xx (6 - 4sqrt2)/(6 - 4sqrt2) = (6^2  + (4sqrt2)^2 - 2 xx 6 4 sqrt2)/((6)^2 - (4sqrt2)^2)`

` (36 + 16 xx 2 - 48sqrt2)/(36 - 16 xx 2)`

`= (36 + 32 - 48sqrt2)/(36 - 32)`

`= (68 - 48sqrt2)/4`

`= 17 - 12sqrt2`

Hence the given expression is simplified with rational denominator to `17 - 12sqrt2`