5.
a)
[tex] \frac{1}{ \sqrt{7} } \times \frac{ \sqrt{7} }{ \sqrt{7} } = \frac{ \sqrt{7} }{7} [/tex]
b)
[tex] \frac{1}{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } = \frac{2 + \sqrt{3} }{4 - 3} = 2 + \sqrt{3} [/tex]
c)
[tex] \frac{7}{4 - \sqrt{2} } \times \frac{4 + \sqrt{2} }{4 + \sqrt{2} } = \frac{7(4 + \sqrt{2} )}{16 - 2} = \frac{7(4 + \sqrt{2}) }{14} = \frac{4 + \sqrt{2} }{2} [/tex]
d)
[tex] \frac{3 - \sqrt{5} }{2 \sqrt{5} - 1} \times \frac{2 \sqrt{5} + 1 }{2 \sqrt{5} + 1} = \frac{(3 - \sqrt{5})(2 \sqrt{5} + 1) }{20 - 1} = \frac{6 \sqrt{5} + 3 - 10 - \sqrt{5} }{19} = \frac{5 \sqrt{5} - 7}{19} [/tex]
6.
[tex](x - \frac{2}{3} )(x + \frac{2}{3}) - {(x - \frac{1}{3} ) }^{2} = - 2x - 4 \frac{1}{9} [/tex]
[tex] {x}^{2} - \frac{4}{9} - ( {x}^{2} - \frac{2}{3} x + \frac{1}{9} ) = - 2x - \frac{37}{9} [/tex]
[tex] {x}^{2} - \frac{4}{9} - {x}^{2} + \frac{2}{3} x - \frac{1}{9} = - 2x - \frac{37}{9} [/tex]
[tex] \frac{2}{3} x + 2x = - \frac{37}{9} + \frac{4}{9} + \frac{1}{9} [/tex]
[tex] \frac{8}{3} x = - \frac{32}{9} [/tex]
[tex]x = - \frac{32}{9} \times \frac{3}{8} = - \frac{4}{3} = - 1 \frac{1}{3} [/tex]
