Question

Simplify the radical expression.

Answer 1

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 $\sqrt{a} + b$ would be$\sqrt{a} - b$.

Applying that information to our fraction shown here, the conjugate of the denominator would be $\sqrt{3} - 4$. We will multiply both the numerator and denominator of our original fraction by this expression to obtain our answer, as shown below.

$\Big(\dfrac{1}{\sqrt{3} + 4}\Big)\Big(\dfrac{\sqrt{3} - 4}{\sqrt{3} - 4}\Big)$

$\dfrac{\sqrt{3} - 4}{3 - 16}$

$\dfrac{4 - \sqrt{3}}{13}$

Our answer is $\boxed{\dfrac{4 - \sqrt{3}}{13}}$.