How do you solve `4x - (x * 3^(1/2)) = 6`?

The idea hee is to isolate `x` on one side of the equation.

To do that, start by using `x` as a common factor for the expression that's on the left-hand side of the equation

`4x - x * 3^(1/2) = 6`

`x * (4 - 3^(1/2)) = 6`

You can rewrite the equation by replacing the fractional exponent by its corresponding radical term

`x * (4 -sqrt(3)) = 6`

DIvide both sides of the equation by `(4 - sqrt(3))` to isolate `x` on the left-hand side of the equation

`( x* color(red)(cancel(color(black)(4-sqrt(3)))))/(color(red)(cancel(color(black)(4-sqrt(3))))) = 6/(4 -sqrt(3))`

`x = 6/(4 - sqrt(3))`

Rationalize the denominator of the fraction by multiplying it by `1 = (4 + sqrt(3))/(4 + sqrt(3))`. The expression `(4 + sqrt(3))` is the conjugate of `(4 - sqrt(3))`. The fraction will thus be equivalent to

`6/(4 - sqrt(3)) * (4 + sqrt(3))/(4 + sqrt(3)) = (6 * (4 + sqrt(3)))/((4 - sqrt(3))(4 + sqrt(3))`

`= (6 * (4 + sqrt(3)))/(4^2 - (sqrt(3))^2)`

`= 6/13 * (4 + sqrt(3))`

Therefore, `x` will be equal to

`x = color(green)(6/13 * (4 + sqrt(3)))`