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?
Accepted Solution
A:
Answer:The value of ABC is 6.Step-by-step explanation:Consider the expression[tex]\frac{1+\sqrt{3}}{1-\sqrt{3}}[/tex]Rationalize the denominator.[tex]\frac{1+\sqrt{3}}{1-\sqrt{3}}\times \frac{1+\sqrt{3}}{1+\sqrt{3}}[/tex][tex]\frac{(1+\sqrt{3})^2}{1^2-(\sqrt{3})^2}[/tex][tex]\frac{1^2+(\sqrt{3})^2+2\sqrt{3}}{1-3}[/tex][tex]\frac{1+3+2\sqrt{3}}{-2}[/tex][tex]\frac{4+2\sqrt{3}}{-2}[/tex][tex]\frac{4}{-2}+\frac{2\sqrt{3}}{-2}[/tex][tex]-2-\sqrt{3}[/tex] ..... (1)The answer in the form [tex]A+B\sqrt{C}[/tex] .... (2)On comparing (1) and (2), we get[tex]A=-2,B=-1,C=3[/tex]We need to find the value of ABC.[tex]ABC=(-2)(-1)(3)=6[/tex]Therefore the value of ABC is 6.