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Sagot :
To simplify the expression [tex]\(\frac{1}{\sqrt{5}+\sqrt{6}-\sqrt{11}}\)[/tex], we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{5}+\sqrt{6}-\sqrt{11}\)[/tex] is [tex]\(\sqrt{5}+\sqrt{6}+\sqrt{11}\)[/tex].
Here are the steps to rationalize the denominator:
1. Multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{1}{\sqrt{5}+\sqrt{6}-\sqrt{11}} \cdot \frac{\sqrt{5}+\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{6}+\sqrt{11}} = \frac{\sqrt{5}+\sqrt{6}+\sqrt{11}}{(\sqrt{5}+\sqrt{6}-\sqrt{11})(\sqrt{5}+\sqrt{6}+\sqrt{11})} \][/tex]
2. Simplify the denominator using the difference of squares:
[tex]\[ (\sqrt{5}+\sqrt{6})^2 - (\sqrt{11})^2 \][/tex]
3. Expand and simplify:
[tex]\[ (\sqrt{5} + \sqrt{6})^2 = (\sqrt{5})^2 + 2\sqrt{5}\sqrt{6} + (\sqrt{6})^2 = 5 + 2\sqrt{30} + 6 = 11 + 2\sqrt{30} \][/tex]
[tex]\[ (\sqrt{11})^2 = 11 \][/tex]
Therefore, the denominator becomes:
[tex]\[ (11 + 2\sqrt{30}) - 11 = 2\sqrt{30} \][/tex]
4. So, the expression now becomes:
[tex]\[ \frac{\sqrt{5}+\sqrt{6}+\sqrt{11}}{2\sqrt{30}} \][/tex]
5. Split the numerator:
[tex]\[ \frac{\sqrt{5}}{2\sqrt{30}} + \frac{\sqrt{6}}{2\sqrt{30}} + \frac{\sqrt{11}}{2\sqrt{30}} \][/tex]
6. Simplify each term:
[tex]\[ \frac{\sqrt{5}}{2\sqrt{30}} = \frac{\sqrt{5}}{2\sqrt{5\cdot6}} = \frac{\sqrt{5}}{2\sqrt{5}\sqrt{6}} = \frac{\sqrt{5}}{2\sqrt{5}\sqrt{6}} = \frac{1}{2\sqrt{6}}, \quad \frac{\sqrt{6}}{2\sqrt{30}} = \frac{\sqrt{6}}{2\sqrt{5\cdot6}} = \frac{\sqrt{6}}{2\sqrt{6}\sqrt{5}} = \frac{1}{2\sqrt{5}}, \quad \frac{\sqrt{11}}{2\sqrt{30}} = \frac{\sqrt{11}}{2\sqrt{5\cdot6}} = \frac{\sqrt{11}}{2\sqrt{5}\sqrt{6}} = \frac{\sqrt{11}}{2\sqrt{30}} \][/tex]
Thus, the simplified format is:
[tex]\[ \frac{1}{\sqrt{5}+\sqrt{6}-\sqrt{11}} = \frac{(\sqrt{5}+\sqrt{6}+\sqrt{11})}{2\sqrt{30}} \][/tex]
In a cleaner format, we can also write it as:
[tex]\[ \boxed{\frac{\sqrt{5} + \sqrt{6} + \sqrt{11}}{2\sqrt{30}}} \][/tex]
Here are the steps to rationalize the denominator:
1. Multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ \frac{1}{\sqrt{5}+\sqrt{6}-\sqrt{11}} \cdot \frac{\sqrt{5}+\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{6}+\sqrt{11}} = \frac{\sqrt{5}+\sqrt{6}+\sqrt{11}}{(\sqrt{5}+\sqrt{6}-\sqrt{11})(\sqrt{5}+\sqrt{6}+\sqrt{11})} \][/tex]
2. Simplify the denominator using the difference of squares:
[tex]\[ (\sqrt{5}+\sqrt{6})^2 - (\sqrt{11})^2 \][/tex]
3. Expand and simplify:
[tex]\[ (\sqrt{5} + \sqrt{6})^2 = (\sqrt{5})^2 + 2\sqrt{5}\sqrt{6} + (\sqrt{6})^2 = 5 + 2\sqrt{30} + 6 = 11 + 2\sqrt{30} \][/tex]
[tex]\[ (\sqrt{11})^2 = 11 \][/tex]
Therefore, the denominator becomes:
[tex]\[ (11 + 2\sqrt{30}) - 11 = 2\sqrt{30} \][/tex]
4. So, the expression now becomes:
[tex]\[ \frac{\sqrt{5}+\sqrt{6}+\sqrt{11}}{2\sqrt{30}} \][/tex]
5. Split the numerator:
[tex]\[ \frac{\sqrt{5}}{2\sqrt{30}} + \frac{\sqrt{6}}{2\sqrt{30}} + \frac{\sqrt{11}}{2\sqrt{30}} \][/tex]
6. Simplify each term:
[tex]\[ \frac{\sqrt{5}}{2\sqrt{30}} = \frac{\sqrt{5}}{2\sqrt{5\cdot6}} = \frac{\sqrt{5}}{2\sqrt{5}\sqrt{6}} = \frac{\sqrt{5}}{2\sqrt{5}\sqrt{6}} = \frac{1}{2\sqrt{6}}, \quad \frac{\sqrt{6}}{2\sqrt{30}} = \frac{\sqrt{6}}{2\sqrt{5\cdot6}} = \frac{\sqrt{6}}{2\sqrt{6}\sqrt{5}} = \frac{1}{2\sqrt{5}}, \quad \frac{\sqrt{11}}{2\sqrt{30}} = \frac{\sqrt{11}}{2\sqrt{5\cdot6}} = \frac{\sqrt{11}}{2\sqrt{5}\sqrt{6}} = \frac{\sqrt{11}}{2\sqrt{30}} \][/tex]
Thus, the simplified format is:
[tex]\[ \frac{1}{\sqrt{5}+\sqrt{6}-\sqrt{11}} = \frac{(\sqrt{5}+\sqrt{6}+\sqrt{11})}{2\sqrt{30}} \][/tex]
In a cleaner format, we can also write it as:
[tex]\[ \boxed{\frac{\sqrt{5} + \sqrt{6} + \sqrt{11}}{2\sqrt{30}}} \][/tex]
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