Question
Rationalise the denominator of the following: $\frac{1}{5+3\sqrt{2}}$
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Answer
If a and b are integers and x is a natural number, then $\big(\text{a}+\text{b}\sqrt{\text{x}}\big)$ and $\big(\text{a}-\text{b}\sqrt{\text{x}}\big)$ are rationalising factor of each other, as $\big(\text{a}+\text{b}\sqrt{\text{x}}\big)\big(\text{a}-\text{b}\sqrt{\text{x}}\big)=\big(\text{a}^2-\text{b}^2\text{x}\big),$ which is rational. Therefore, we have, $=\frac{1}{\big(5+3\sqrt{2}\big)}=\frac{1}{5+3\sqrt{2}}\times\frac{5-3\sqrt{2}}{5-3\sqrt{2}}$ $=\frac{5-3\sqrt{2}}{(5)^2-\big(3\sqrt{2}\big)^2}=\frac{5-3\sqrt{2}}{25-18}=\Big(\frac{5-3\sqrt{2}}{7}\Big)$
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