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

Answer:

The value of ABC is 6.

Step-by-step explanation:

Consider the expression

$\frac{1+\sqrt{3}}{1-\sqrt{3}}$

Rationalize the denominator.

$\frac{1+\sqrt{3}}{1-\sqrt{3}}\times \frac{1+\sqrt{3}}{1+\sqrt{3}}$

$\frac{(1+\sqrt{3})^2}{1^2-(\sqrt{3})^2}$

$\frac{1^2+(\sqrt{3})^2+2\sqrt{3}}{1-3}$

$\frac{1+3+2\sqrt{3}}{-2}$

$\frac{4+2\sqrt{3}}{-2}$

$\frac{4}{-2}+\frac{2\sqrt{3}}{-2}$

$-2-\sqrt{3}$              ..... (1)

The answer in the form

$A+B\sqrt{C}$           .... (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.