Answer
$\dfrac{12-3\sqrt{5}}{11}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\dfrac{3}{4+\sqrt{5}}
,$ multiply the numerator and the denominator by the conjugate of the denominator. Then use special products to multiply the result.
$\bf{\text{Solution Details:}}$
Multiplying the numerator and the denominator of the given expression by the conjugate of the denominator, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{3}{4+\sqrt{5}}\cdot\dfrac{4-\sqrt{5}}{4-\sqrt{5}}
\\\\=
\dfrac{3(4-\sqrt{5})}{(4+\sqrt{5})(4-\sqrt{5})}
.\end{array}
Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\dfrac{3(4-\sqrt{5})}{(4)^2-(\sqrt{5})^2}
\\\\=
\dfrac{3(4-\sqrt{5})}{16-5}
\\\\=
\dfrac{3(4-\sqrt{5})}{11}
.\end{array}
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{3(4)-3(\sqrt{5})}{11}
\\\\=
\dfrac{12-3\sqrt{5}}{11}
.\end{array}
