Number system practice

Rationalise the denominator

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

Video Explanation

Explanatory Answer

\\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 Rationalisation

= \\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 Rationalisation

\\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}\\)