To simplify the given expression we need to remove the radical from the denominator by rationalizing it.
To rationalize a radical expression first step is multiply both numerator and denominator of the expression by the conjugate of it's denominator.
So, conjugate of 6+√3 = 6-√3.
Hence,
[tex] \frac{1}{6+\sqrt{3}} *\frac{6-\sqrt{3}}{6-\sqrt{3}} [/tex]
=[tex] \frac{6-\sqrt{3}}{(6+\sqrt{3})(6-\sqrt{3})} [/tex]
=[tex] \frac{6-\sqrt{3}}{6^2-\sqrt{3}^2} [/tex] Since (a+b)(a-b)= a^2-b^2
=[tex] \frac{6-\sqrt{3}}{36-3} [/tex]
=[tex] \frac{6-\sqrt{3}}{33} [/tex]
So, the answer is [tex] \frac{6-\sqrt{3}}{33} [/tex]
