Question

Rationalize the denominator of \frac{1 \sqrt{3}}{1-\sqrt{3}}. When you write your answer in the form A B\sqrt{C}, where A, B, and C are integers, what is ABC?

Answer 1

The value of ABC is 6.

Explanation:

Consider the expression

`(1+√(3))/(1-√(3))`

Rationalize the denominator.

`(1+√(3))/(1-√(3))* (1+√(3))/(1+√(3))`

`((1+√(3))^2)/(1^2-(√(3))^2)`

`(1^2+(√(3))^2+2√(3))/(1-3)`

`(1+3+2√(3))/(-2)`

`(4+2√(3))/(-2)`

`(4)/(-2)+(2√(3))/(-2)`

`-2-√(3)` ..... (1)

The answer in the form

`A+B√(C)` .... (2)

On comparing (1) and (2), we get

`A=-2,B=-1,C=3`

We need to find the value of ABC.

`ABC=(-2)(-1)(3)=6`

Therefore the value of ABC is 6.