# Solve for: (4+sqrt(18))/(4-sqrt(18))

## Expression: $\frac{ 4+\sqrt{ 18 } }{ 4-\sqrt{ 18 } }$

$\frac{ 4+3\sqrt{ 2 } }{ 4-\sqrt{ 18 } }$

$\frac{ 4+3\sqrt{ 2 } }{ 4-3\sqrt{ 2 } }$

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

$\frac{ 16+24\sqrt{ 2 }+18 }{ 16-18 }$

$\frac{ 34+24\sqrt{ 2 } }{ 16-18 }$

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*}

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