Question

a = 3+ 2sqrt( 2);a ^ ( 1/2) - a ^ ( - 1/ 2) = ?

Answer 1

Given...
$a = 3 + 2 \sqrt{2} \\ we \: need \: to \: find \: {a}^{ \frac{1}{2} } - {a}^{ - \frac{1}{2} } \\ now \: put \: value \: of \: a... \\ = {(3 + 2 \sqrt{2}) }^{ \frac{1}{2} } - {(3 + 2 \sqrt{2}) }^{ \frac{ - 1}{2} } \\ = {( { \sqrt{2} }^{2} + 1 + 2 \sqrt{2}) }^{ \frac{1}{2} } - {( { \sqrt{2} }^{2} + 1 + 2 \sqrt{2}) }^{ \frac{ - 1}{2} } \\ = {( \sqrt{2} + 1)}^{2 \times \frac{1}{2} } - {( \sqrt{2} + 1)}^{2 \times \frac{ - 1}{2} } \\ = ( \sqrt{2} + 1) - {( \sqrt{2} + 1)}^{ - 1} \\ = ( \sqrt{2} + 1) - \frac{1}{( \sqrt{2} + 1)} \: \: \: \: multiply \: and \: divide \: by \: \sqrt{2} - 1 \\ = ( \sqrt{2} + 1) - ( \sqrt{2} - 1) \\ = \sqrt{2} + 1 - \sqrt{2} + 1 \\ = 2 \\ \\ \\ plz \: follow \: me \: if \: this \: answer \: is \: helpful$