Answer :
Let's rationalize the denominator of each given expression step-by-step:
### (i) [tex]\(\frac{2}{\sqrt{3} - \sqrt{5}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{2}{\sqrt{3} - \sqrt{5}} \times \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} = \frac{2 (\sqrt{3} + \sqrt{5})}{(\sqrt{3} - \sqrt{5}) (\sqrt{3} + \sqrt{5})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5}) = (\sqrt{3})^2 - (\sqrt{5})^2 = 3 - 5 = -2 \][/tex]
Thus, we have:
[tex]\[ \frac{2 (\sqrt{3} + \sqrt{5})}{-2} = - (\sqrt{3} + \sqrt{5}) \][/tex]
### (ii) [tex]\(\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}\)[/tex]
Similarly, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \times \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} = \frac{(\sqrt{3} + \sqrt{2})^2}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \][/tex]
The numerator simplifies as follows:
[tex]\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{6} = 5 + 2\sqrt{6} \][/tex]
Thus, we have:
[tex]\[ \frac{5 + 2\sqrt{6}}{1} = 5 + 2\sqrt{6} \][/tex]
### (iii) [tex]\(\frac{6}{\sqrt{5} + \sqrt{2}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{6}{\sqrt{5} + \sqrt{2}} \times \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{6 (\sqrt{5} - \sqrt{2})}{(\sqrt{5} + \sqrt{2}) (\sqrt{5} - \sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3 \][/tex]
Thus, we have:
[tex]\[ \frac{6 (\sqrt{5} - \sqrt{2})}{3} = 2 (\sqrt{5} - \sqrt{2}) = 2\sqrt{5} - 2\sqrt{2} \][/tex]
### (iv) [tex]\(\frac{1}{8 + 5\sqrt{2}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{1}{8 + 5\sqrt{2}} \times \frac{8 - 5\sqrt{2}}{8 - 5\sqrt{2}} = \frac{8 - 5\sqrt{2}}{(8 + 5\sqrt{2})(8 - 5\sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (8 + 5\sqrt{2})(8 - 5\sqrt{2}) = 8^2 - (5\sqrt{2})^2 = 64 - 25 \times 2 = 64 - 50 = 14 \][/tex]
Thus, we have:
[tex]\[ \frac{8 - 5\sqrt{2}}{14} = \frac{8}{14} - \frac{5\sqrt{2}}{14} = \frac{4}{7} - \frac{5\sqrt{2}}{14} \][/tex]
### (v) [tex]\(\frac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}} \times \frac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} = \frac{(3 - 2\sqrt{2})^2}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \][/tex]
The numerator simplifies as follows:
[tex]\[ (3 - 2\sqrt{2})^2 = 3^2 - 2 \times 3 \times 2\sqrt{2} + (2\sqrt{2})^2 = 9 - 12\sqrt{2} + 8 = 17 - 12\sqrt{2} \][/tex]
Thus, we have:
[tex]\[ \frac{17 - 12\sqrt{2}}{1} = 17 - 12\sqrt{2} \][/tex]
### (vi) [tex]\(\frac{\sqrt{3} - 1}{\sqrt{3} + 1}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \times \frac{\sqrt{3} - 1}{\sqrt{3} - 1} = \frac{(\sqrt{3} - 1)^2}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2 \][/tex]
The numerator simplifies as follows:
[tex]\[ (\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2\sqrt{3} + 1 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \][/tex]
Thus, we have:
[tex]\[ \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \][/tex]
### (vii) [tex]\(\frac{4}{\sqrt{7} + \sqrt{3}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{4}{\sqrt{7} + \sqrt{3}} \times \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}} = \frac{4 (\sqrt{7} - \sqrt{3})}{(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = (\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4 \][/tex]
Thus, we have:
[tex]\[ \frac{4 (\sqrt{7} - \sqrt{3})}{4} = \sqrt{7} - \sqrt{3} \][/tex]
### (viii) [tex]\(\frac{1}{5 + 3\sqrt{2}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{1}{5 + 3\sqrt{2}} \times \frac{5 - 3\sqrt{2}}{5 - 3\sqrt{2}} = \frac{5 - 3\sqrt{2}}{(5 + 3\sqrt{2})(5 - 3\sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (5 + 3\sqrt{2})(5 - 3\sqrt{2}) = 5^2 - (3\sqrt{2})^2 = 25 - 9 \times 2 = 25 - 18 = 7 \][/tex]
Thus, we have:
[tex]\[ \frac{5 - 3\sqrt{2}}{7} \][/tex]
This completes the detailed, step-by-step rationalization of the denominators for each given expression.
### (i) [tex]\(\frac{2}{\sqrt{3} - \sqrt{5}}\)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{2}{\sqrt{3} - \sqrt{5}} \times \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} = \frac{2 (\sqrt{3} + \sqrt{5})}{(\sqrt{3} - \sqrt{5}) (\sqrt{3} + \sqrt{5})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5}) = (\sqrt{3})^2 - (\sqrt{5})^2 = 3 - 5 = -2 \][/tex]
Thus, we have:
[tex]\[ \frac{2 (\sqrt{3} + \sqrt{5})}{-2} = - (\sqrt{3} + \sqrt{5}) \][/tex]
### (ii) [tex]\(\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}\)[/tex]
Similarly, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \times \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} = \frac{(\sqrt{3} + \sqrt{2})^2}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \][/tex]
The numerator simplifies as follows:
[tex]\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{6} = 5 + 2\sqrt{6} \][/tex]
Thus, we have:
[tex]\[ \frac{5 + 2\sqrt{6}}{1} = 5 + 2\sqrt{6} \][/tex]
### (iii) [tex]\(\frac{6}{\sqrt{5} + \sqrt{2}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{6}{\sqrt{5} + \sqrt{2}} \times \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{6 (\sqrt{5} - \sqrt{2})}{(\sqrt{5} + \sqrt{2}) (\sqrt{5} - \sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3 \][/tex]
Thus, we have:
[tex]\[ \frac{6 (\sqrt{5} - \sqrt{2})}{3} = 2 (\sqrt{5} - \sqrt{2}) = 2\sqrt{5} - 2\sqrt{2} \][/tex]
### (iv) [tex]\(\frac{1}{8 + 5\sqrt{2}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{1}{8 + 5\sqrt{2}} \times \frac{8 - 5\sqrt{2}}{8 - 5\sqrt{2}} = \frac{8 - 5\sqrt{2}}{(8 + 5\sqrt{2})(8 - 5\sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (8 + 5\sqrt{2})(8 - 5\sqrt{2}) = 8^2 - (5\sqrt{2})^2 = 64 - 25 \times 2 = 64 - 50 = 14 \][/tex]
Thus, we have:
[tex]\[ \frac{8 - 5\sqrt{2}}{14} = \frac{8}{14} - \frac{5\sqrt{2}}{14} = \frac{4}{7} - \frac{5\sqrt{2}}{14} \][/tex]
### (v) [tex]\(\frac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}} \times \frac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} = \frac{(3 - 2\sqrt{2})^2}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \][/tex]
The numerator simplifies as follows:
[tex]\[ (3 - 2\sqrt{2})^2 = 3^2 - 2 \times 3 \times 2\sqrt{2} + (2\sqrt{2})^2 = 9 - 12\sqrt{2} + 8 = 17 - 12\sqrt{2} \][/tex]
Thus, we have:
[tex]\[ \frac{17 - 12\sqrt{2}}{1} = 17 - 12\sqrt{2} \][/tex]
### (vi) [tex]\(\frac{\sqrt{3} - 1}{\sqrt{3} + 1}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \times \frac{\sqrt{3} - 1}{\sqrt{3} - 1} = \frac{(\sqrt{3} - 1)^2}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2 \][/tex]
The numerator simplifies as follows:
[tex]\[ (\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2\sqrt{3} + 1 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \][/tex]
Thus, we have:
[tex]\[ \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \][/tex]
### (vii) [tex]\(\frac{4}{\sqrt{7} + \sqrt{3}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{4}{\sqrt{7} + \sqrt{3}} \times \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}} = \frac{4 (\sqrt{7} - \sqrt{3})}{(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = (\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4 \][/tex]
Thus, we have:
[tex]\[ \frac{4 (\sqrt{7} - \sqrt{3})}{4} = \sqrt{7} - \sqrt{3} \][/tex]
### (viii) [tex]\(\frac{1}{5 + 3\sqrt{2}}\)[/tex]
We multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{1}{5 + 3\sqrt{2}} \times \frac{5 - 3\sqrt{2}}{5 - 3\sqrt{2}} = \frac{5 - 3\sqrt{2}}{(5 + 3\sqrt{2})(5 - 3\sqrt{2})} \][/tex]
The denominator simplifies as follows:
[tex]\[ (5 + 3\sqrt{2})(5 - 3\sqrt{2}) = 5^2 - (3\sqrt{2})^2 = 25 - 9 \times 2 = 25 - 18 = 7 \][/tex]
Thus, we have:
[tex]\[ \frac{5 - 3\sqrt{2}}{7} \][/tex]
This completes the detailed, step-by-step rationalization of the denominators for each given expression.
