ANSWER
[tex] \frac{11}{9} [/tex]
EXPLANATION
The conjugate of
[tex] - \frac{ \sqrt{2} }{3} + i[/tex]
is
[tex] - \frac{ \sqrt{2} }{3} - i[/tex]
The product is
[tex]( - \frac{ \sqrt{2} }{3} + i)( - \frac{ \sqrt{2} }{3} - i)[/tex]
We apply difference of two squares to obtain,
[tex] = {( - \frac{ \sqrt{2} }{3} )}^{2} - {i}^{2} )[/tex]
Recall that,
[tex] {i}^{2} = - 1[/tex]
This implies that,
[tex] = ( \frac{2}{9} - ( - 1))[/tex]
[tex] = \frac{2}{9} + 1[/tex]
[tex] = \frac{11}{9} [/tex]
