Answer:
[tex]13 + 7\sqrt{3}[/tex]
Step-by-step explanation:
Given
[tex]\frac{5 + \sqrt{3}}{2 - \sqrt{3}}[/tex]
Required
Rationalize
To rationalize, we have to multiply the numerator and the denominator by [tex]{2 + \sqrt{3}}[/tex]
This gives
[tex]\frac{5 + \sqrt{3}}{2 - \sqrt{3}} * \frac{2 + \sqrt{3}}{2 + \sqrt{3}}[/tex]
[tex]\frac{(5 + \sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}[/tex]
Expand
[tex]\frac{5(2 + \sqrt{3})+\sqrt{3}(2 + \sqrt{3})}{2(2 + \sqrt{3}) - \sqrt{3}(2 + \sqrt{3})}[/tex]
Open the brackets
[tex]\frac{10 + 5\sqrt{3}+2\sqrt{3} + (\sqrt{3})^2}{4 + 2\sqrt{3} - 2\sqrt{3} - (\sqrt{3})^2}[/tex]
[tex]\frac{10 + 5\sqrt{3}+2\sqrt{3} + 3}{4 + 2\sqrt{3} - 2\sqrt{3} - 3}[/tex]
Collect like terms for ease of arithmetic operations
[tex]\frac{10 + 3 + 5\sqrt{3}+2\sqrt{3}}{4 -3 + 2\sqrt{3} - 2\sqrt{3}}[/tex]
[tex]\frac{13 + 7\sqrt{3}}{1 }[/tex]
[tex]13 + 7\sqrt{3}[/tex]
Hence,
[tex]\frac{5 + \sqrt{3}}{2 - \sqrt{3}}[/tex] is equivalent to [tex]13 + 7\sqrt{3}[/tex], after rationalizing the denominator
