To simplify this fraction, multiply the entire fraction by the conjugate of the denominator. The conjugate of a square root and a number being added to it would be the number subtracted from the square root. In other words, the conjugate of [tex] \sqrt{a} + b [/tex] would be[tex] \sqrt{a} - b [/tex].
Applying that information to our fraction shown here, the conjugate of the denominator would be [tex] \sqrt{3} - 4 [/tex]. We will multiply both the numerator and denominator of our original fraction by this expression to obtain our answer, as shown below.
[tex] \Big(\dfrac{1}{\sqrt{3} + 4}\Big)\Big(\dfrac{\sqrt{3} - 4}{\sqrt{3} - 4}\Big) [/tex]
[tex] \dfrac{\sqrt{3} - 4}{3 - 16} [/tex]
[tex] \dfrac{4 - \sqrt{3}}{13} [/tex]
Our answer is [tex] \boxed{\dfrac{4 - \sqrt{3}}{13}} [/tex].
