Question 10: Rationalize the denominator of \\frac{1}{9 + {\sqrt{5} + \sqrt{6}}}\\)

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\\frac{1}{9 + \sqrt{5} + \sqrt{6}})

Rationalising the denominator of this question has to be done in two stages.

In each of the stages, we will be multiplying the numerator and the denominator of the fraction with the conjugate of the denominator.

**Stage 1**: The conjucate of (9 + \\sqrt{5}) + \\sqrt{6})) is (9 - (\\sqrt{5}) + \\sqrt{6})))

= \\frac{1}{9 + \sqrt{5} + \sqrt{6}}) x \\frac{9 - (\sqrt{5} + \sqrt{6})}{9 - (\sqrt{5} + \sqrt{6})})

= \\frac{9 - (\sqrt{5} + \sqrt{6})}{9^2 - (\sqrt{5} + \sqrt{6})^2})

= \\frac{9 - (\sqrt{5} + \sqrt{6})}{81 - (5 + 6 + 2\sqrt{30})})

= \\frac{9 - ({\sqrt{5} + \sqrt{6}})}{70 - 2{\sqrt{30}}}\\)

**Stage 2**: The conjucate of (70 - 2\\sqrt{30})) is (70 + 2\\sqrt{30}))

\\frac{9 - (\sqrt{5} + \sqrt{6})}{70 - 2\sqrt{30}}) x \\frac{70 + 2\sqrt{30}}{70 + 2\sqrt{30}})

= \\frac{630 + 18\sqrt{30} - 70\sqrt{5} - 10\sqrt{6} - 70\sqrt{6} - 12\sqrt{5}} {4900 - 120}\\)

= \\frac{630 + 18\sqrt{30} - 82\sqrt{5} - 80\sqrt{6}} {4780})

Class 9 Maths

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