Answer :
Sure, let's simplify each of the given expressions by rationalizing their denominators.
### Part (i) Rationalizing the denominator of \( \frac{1}{\sqrt{6}} \)
To rationalize the denominator of the expression \( \frac{1}{\sqrt{6}} \), we need to eliminate the square root from the denominator. We do this by multiplying both the numerator and the denominator by \( \sqrt{6} \).
[tex]\[ \frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{6}}{6} \][/tex]
So the simplified form is:
[tex]\[ \frac{\sqrt{6}}{6} \approx 0.4082 \][/tex]
### Part (ii) Rationalizing the denominator of \( \frac{1}{\sqrt{5} + \sqrt{2}} \)
To rationalize the denominator of \( \frac{1}{\sqrt{5} + \sqrt{2}} \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{5} - \sqrt{2} \).
[tex]\[ \frac{1}{\sqrt{5} + \sqrt{2}} \cdot \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{\sqrt{5} - \sqrt{2}}{(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})} \][/tex]
In the denominator, we use the difference of squares formula \( a^2 - b^2 \):
[tex]\[ (\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3 \][/tex]
Thus, the expression simplifies to:
[tex]\[ \frac{\sqrt{5} - \sqrt{2}}{3} \][/tex]
And approximately:
[tex]\[ \frac{\sqrt{5} - \sqrt{2}}{3} \approx 0.2740 \][/tex]
To summarize:
1. \( \frac{1}{\sqrt{6}} \) simplifies to \( \frac{\sqrt{6}}{6} \approx 0.4082 \).
2. \( \frac{1}{\sqrt{5} + \sqrt{2}} \) simplifies to \( \frac{\sqrt{5} - \sqrt{2}}{3} \approx 0.2740 \).
These are the simplified forms of the given expressions with their denominators rationalized.
### Part (i) Rationalizing the denominator of \( \frac{1}{\sqrt{6}} \)
To rationalize the denominator of the expression \( \frac{1}{\sqrt{6}} \), we need to eliminate the square root from the denominator. We do this by multiplying both the numerator and the denominator by \( \sqrt{6} \).
[tex]\[ \frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{6}}{6} \][/tex]
So the simplified form is:
[tex]\[ \frac{\sqrt{6}}{6} \approx 0.4082 \][/tex]
### Part (ii) Rationalizing the denominator of \( \frac{1}{\sqrt{5} + \sqrt{2}} \)
To rationalize the denominator of \( \frac{1}{\sqrt{5} + \sqrt{2}} \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{5} - \sqrt{2} \).
[tex]\[ \frac{1}{\sqrt{5} + \sqrt{2}} \cdot \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{\sqrt{5} - \sqrt{2}}{(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})} \][/tex]
In the denominator, we use the difference of squares formula \( a^2 - b^2 \):
[tex]\[ (\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3 \][/tex]
Thus, the expression simplifies to:
[tex]\[ \frac{\sqrt{5} - \sqrt{2}}{3} \][/tex]
And approximately:
[tex]\[ \frac{\sqrt{5} - \sqrt{2}}{3} \approx 0.2740 \][/tex]
To summarize:
1. \( \frac{1}{\sqrt{6}} \) simplifies to \( \frac{\sqrt{6}}{6} \approx 0.4082 \).
2. \( \frac{1}{\sqrt{5} + \sqrt{2}} \) simplifies to \( \frac{\sqrt{5} - \sqrt{2}}{3} \approx 0.2740 \).
These are the simplified forms of the given expressions with their denominators rationalized.
