Question

Calculate the expression step by step `(root3(2+sqrt3)\timessqrt(7-4\timessqrt3))`? Answer should be `(root3(7-4\timessqrt3))` and I just want to know HOW?

`root3(2+sqrt3)\timessqrt(7-4\timessqrt3)`

Answer 1

Please see below.

Explanation:

`2+sqrt3`

= `((2+sqrt3)(2-sqrt3))/(2-sqrt3)`

= `(2^2-(sqrt3)^2)/(2-sqrt3)`

= `1/(2-sqrt3)`

= `1/sqrt((2-sqrt3)^2)`

= `1/sqrt(4+(sqrt3)^2-4sqrt3)`

= `1/sqrt(7-4sqrt3)`

= `(7-4sqrt3)^(-1/2)`

Hence `root(3)(2+sqrt3)xxsqrt(7-4sqrt3)`

= `(2+sqrt3)^(1/3)xx(7-4sqrt3)^(1/2)`

= `((7-4sqrt3)^(-1/2))^(1/3)xx(7-4sqrt3)^(1/2)`

= `(7-4sqrt3)^(-1/6+1/2)`

= `(7-4sqrt3)^(1/3)`

= `root(3)(7-4sqrt3)`