$\frac{ 4+3\sqrt{ 2 } }{ 4-\sqrt{ 18 } }$
Simplify the radical expression$\frac{ 4+3\sqrt{ 2 } }{ 4-3\sqrt{ 2 } }$
Multiply the fraction by $\frac{ 4+3\sqrt{ 2 } }{ 4+3\sqrt{ 2 } }$$\frac{ 4+3\sqrt{ 2 } }{ 4-3\sqrt{ 2 } } \times \frac{ 4+3\sqrt{ 2 } }{ 4+3\sqrt{ 2 } }$
To multiply the fractions, multiply the numerators and denominators separately$\frac{ \left( 4+3\sqrt{ 2 } \right) \times \left( 4+3\sqrt{ 2 } \right) }{ \left( 4-3\sqrt{ 2 } \right) \times \left( 4+3\sqrt{ 2 } \right) }$
The factor $4+3\sqrt{ 2 }$ repeats $2$ times, so the base is $4+3\sqrt{ 2 }$ and the exponent is $2$$\frac{ {\left( 4+3\sqrt{ 2 } \right)}^{2} }{ \left( 4-3\sqrt{ 2 } \right) \times \left( 4+3\sqrt{ 2 } \right) }$
Use $\left( a-b \right)\left( a+b \right)={a}^{2}-{b}^{2}$ to simplify the product$\frac{ {\left( 4+3\sqrt{ 2 } \right)}^{2} }{ 16-9 \times 2 }$
Use ${\left( a+b \right)}^{2}={a}^{2}+2ab+{b}^{2}$ to expand the expression$\frac{ 16+24\sqrt{ 2 }+18 }{ 16-9 \times 2 }$
Multiply the numbers$\frac{ 16+24\sqrt{ 2 }+18 }{ 16-18 }$
Add the numbers$\frac{ 34+24\sqrt{ 2 } }{ 16-18 }$
Calculate the difference$\frac{ 34+24\sqrt{ 2 } }{ -2 }$
Use $\frac{ -a }{ b }=\frac{ a }{ -b }=-\frac{ a }{ b }$ to rewrite the fraction$-\frac{ 34+24\sqrt{ 2 } }{ 2 }$
Factor out $2$ from the expression$-\frac{ 2\left( 17+12\sqrt{ 2 } \right) }{ 2 }$
Cancel out the common factor $2$$-\left( 17+12\sqrt{ 2 } \right)$
When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses$\begin{align*}&-17-12\sqrt{ 2 } \\&\approx-33.97056\end{align*}$