# If  x = 3 +2sqrt(2) Then what is x ^ (1/2) - x ^ (-1/2)?

Jul 23, 2017

${x}^{\frac{1}{2}} - {x}^{- \frac{1}{2}} = 2$

#### Explanation:

Note that:

${\left(a + b \sqrt{2}\right)}^{2} = \left({a}^{2} + 2 {b}^{2}\right) + 2 a b \sqrt{2}$

Comparing the right hand expression with $3 + 2 \sqrt{2}$, note that it will match if we put $a = b = 1$, so:

${\left(1 + \sqrt{2}\right)}^{2} = 3 + 2 \sqrt{2}$

So $1 + \sqrt{2}$ is a square root of $3 + 2 \sqrt{2}$. Since it is positive, it is the principal square root, so:

${\left(3 + 2 \sqrt{2}\right)}^{\frac{1}{2}} = 1 + \sqrt{2}$

So with $x = 3 + 2 \sqrt{2}$ we find:

${x}^{\frac{1}{2}} - {x}^{- \frac{1}{2}} = \left(1 + \sqrt{2}\right) - \frac{1}{1 + \sqrt{2}}$

$\textcolor{w h i t e}{{x}^{\frac{1}{2}} - {x}^{- \frac{1}{2}}} = \left(1 + \sqrt{2}\right) - \frac{1 - \sqrt{2}}{\left(1 + \sqrt{2}\right) \left(1 - \sqrt{2}\right)}$

$\textcolor{w h i t e}{{x}^{\frac{1}{2}} - {x}^{- \frac{1}{2}}} = \left(1 + \sqrt{2}\right) - \frac{1 - \sqrt{2}}{1 - 2}$

$\textcolor{w h i t e}{{x}^{\frac{1}{2}} - {x}^{- \frac{1}{2}}} = \left(1 + \textcolor{red}{\cancel{\textcolor{b l a c k}{\sqrt{2}}}}\right) + \left(1 - \textcolor{red}{\cancel{\textcolor{b l a c k}{\sqrt{2}}}}\right)$

$\textcolor{w h i t e}{{x}^{\frac{1}{2}} - {x}^{- \frac{1}{2}}} = 2$