Answer
$\dfrac{3\sqrt{2}+2\sqrt{7}}{10}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\dfrac{-1}{3\sqrt{2}-2\sqrt{7}}
,$ multiply the numerator and the denominator by the conjugate of the denominator. Then use special products to simplify 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{-1}{3\sqrt{2}-2\sqrt{7}} \cdot\dfrac{3\sqrt{2}+2\sqrt{7}}{3\sqrt{2}+2\sqrt{7}}
\\\\=
\dfrac{-1(3\sqrt{2}+2\sqrt{7})}{(3\sqrt{2}-2\sqrt{7})(3\sqrt{2}+2\sqrt{7})}
.\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{-1(3\sqrt{2}+2\sqrt{7})}{(3\sqrt{2})^2-(2\sqrt{7})^2}
\\\\=
\dfrac{-1(3\sqrt{2}+2\sqrt{7})}{9(2)-4(7)}
\\\\=
\dfrac{-1(3\sqrt{2}+2\sqrt{7})}{18-28}
\\\\=
\dfrac{-1(3\sqrt{2}+2\sqrt{7})}{-10}
\\\\=
\dfrac{1(3\sqrt{2}+2\sqrt{7})}{10}
.\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{1(3\sqrt{2})+1(2\sqrt{7})}{10}
\\\\=
\dfrac{3\sqrt{2}+2\sqrt{7}}{10}
.\end{array}
