[tex]$ \frac{5+\sqrt2}{23}[/tex]
Solution:
Given expression is [tex]\frac{1}{5-\sqrt{2}}[/tex].
Rationalise the denominator means removing the root terms in the denominator.
To rationalise the denominator:
[tex]$\frac{1}{5-\sqrt{2}}[/tex]
Multiply the numerator and denominator by the conjugate [tex]5+\sqrt{2}[/tex].
[tex]$\Rightarrow\frac{1}{5-\sqrt{2}}\times\frac{5+\sqrt{2}}{5+\sqrt{2}}[/tex]
[tex]$\Rightarrow \frac{1(5+\sqrt2)}{(5-\sqrt2)(5+\sqrt2)}[/tex]
Use the algebraic identity [tex](a-b)(a+b)=a^{2}-b^{2}[/tex] in the denominator.
[tex]$\Rightarrow \frac{5+\sqrt2}{5^{2}-(\sqrt{2})^{2}}[/tex]
[tex]$\Rightarrow \frac{5+\sqrt2}{25-2}[/tex]
[tex]$\Rightarrow \frac{5+\sqrt2}{23}[/tex]
Hence the denominator is rationalised.
