Rasionalkan pecahan berikut.
[tex] 1.) \: \bf \frac{4}{10 - \sqrt{6} } = . \: . \: .[/tex]
[tex]\:[/tex]
[tex]2.) \: \bf \frac{4 - \sqrt{5} }{2 \sqrt{3} + \sqrt{6} } = . \: . \: .[/tex]
[tex] \: [/tex]
[tex] \to \sf \underline{Rules : }[/tex]
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Pembahasan :
[tex]1.) \: \frac{4}{10 - \sqrt{6} } = ...[/tex]
Penjelasan :
[tex]\frac{4}{10 - \sqrt{6} } [/tex]
[tex] = \frac{4}{10 - \sqrt{6} } \times \frac{10 + \sqrt{6} }{10 + \sqrt{6} } [/tex]
[tex] = \frac{4(10 + \sqrt{6}) }{(10 - \sqrt{6} ) \times (10 + \sqrt{6} )} [/tex]
[tex] = \frac{4(10 + \sqrt{6}) }{10 {}^{2} - \sqrt{6} {}^{2} } = \frac{4(10 + \sqrt{6} )}{100 - 6} [/tex]
[tex] = \frac{4(10 + \sqrt{6} )}{94} = \frac{2(10 + \sqrt{6} )}{47} [/tex]
[tex] = \frac{20 + 2 \sqrt{6} }{47} [/tex]
***
[tex]2.) \: \frac{4 - \sqrt{5} }{2 \sqrt{3} + \sqrt{6} } = ...[/tex]
Penyelesaian :
[tex] \frac{4 - \sqrt{5} }{2 \sqrt{3} + \sqrt{6} } [/tex]
[tex] = \frac{4 - \sqrt{5} }{2 \sqrt{3} + \sqrt{6} } \times \frac{2 \sqrt{3} - \sqrt{6} }{2 \sqrt{3} - \sqrt{6} } [/tex]
[tex] = \frac{(4 - \sqrt{5}) \times (2 \sqrt{3} - \sqrt{6} ) }{(2 \sqrt{3} + \sqrt{6}) \times (2 \sqrt{3} - \sqrt{6}) } [/tex]
[tex] = \frac{4 \times 2 \sqrt{3} - 4 \sqrt{6} - \sqrt{5} \times 2 \sqrt{3} - \sqrt{5} \times ( - \sqrt{6} ) }{(2 \sqrt{3} ) {}^{2} - \sqrt{6} {}^{2} } [/tex]
[tex] = \frac{8 \sqrt{3} - 4 \sqrt{6} - 2 \sqrt{15 + \sqrt{30} } }{4 \times 3 - 6} [/tex]
[tex] = \frac{8 \sqrt{3} - 4 \sqrt{6} - 2 \sqrt{15 + \sqrt{30} } }{ 6} [/tex]
[tex] \sf\frac{4}{10 - \sqrt{6} }[/tex]
[tex] \sf = \frac{4}{10 - \sqrt{6} } \times \frac{10 + \sqrt{6} }{10 + \sqrt{6} } [/tex]
[tex] \sf = \frac{4(10 + \sqrt{6}) }{100 - 6} [/tex]
[tex] \sf = \frac{4(10 + \sqrt{6}) }{94} [/tex]
[tex] \sf = \frac{2(10 + \sqrt{6}) }{47} [/tex]
[tex] \sf = \frac{20 + 2\sqrt{6} }{47} [/tex]
...
[tex]\sf \frac{4 - \sqrt{5} }{2 \sqrt{3} + \sqrt{6} }[/tex]
[tex]\sf = \frac{4 - \sqrt{5} }{2 \sqrt{3} + \sqrt{6} } \times \frac{2 \sqrt{3} - \sqrt{6} }{2 \sqrt{3} - \sqrt{6} } [/tex]
[tex]\sf = \frac{(4 - \sqrt{5})(2 \sqrt{3} - \sqrt{6} )}{12 - 6 }[/tex]
[tex]\sf = \frac{(4 - \sqrt{5})(2 \sqrt{3} - \sqrt{6} )}{6 }[/tex]
[tex]\sf = \frac{8 \sqrt{3} - 4 \sqrt{6} - 2 \sqrt{15} + \sqrt{30} }{6 }[/tex]
[answer.2.content]
