Merasionalkan Pecahan

Merasionalkan adalah mengubah bilangan pecahan berpenyebut akar menjadi berpenyebut bilangan bulat.

Beberapa cara merasionalkan bentuk akar.

Mengalikan dengan akar penyebut

1a=1a×aa=aa\large\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{a}}\times \frac{\sqrt{a}}{\sqrt{a}}=\frac{\sqrt{a}}{a}Contoh 1:62=62×22=622=32\large\frac{6}{\sqrt{2}}=\frac{6}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=\normalsize 3\sqrt{2}Contoh 2:35−105=35−105×55=35.5−1055\large\frac{3\sqrt{5}-10}{\sqrt{5}}=\frac{3\sqrt{5}-10}{\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}}=\frac{3\sqrt{5}.\sqrt{5}-10\sqrt{5}}{5}=3×5−1055\large\hspace8.5em =\frac{3\times 5 - 10\sqrt{5}}{5}=5(3−25)5\large\hspace8.5em =\frac{5(3 - 2\sqrt{5})}{5}=3−25\large\hspace8.5em =\normalsize 3-2\sqrt{5}

Mengalikan dengan akar sekawan

1a±b=1a±b×a∓ba∓b=a∓ba−b\large\frac{1}{\sqrt{a}\pm \sqrt{b}}=\frac{1}{\sqrt{a}\pm \sqrt{b}}\times \frac{\sqrt{a}\mp \sqrt{b}}{\sqrt{a}\mp \sqrt{b}}=\frac{\sqrt{a}\mp \sqrt{b}}{a-b}Contoh 1:32+1=32+1×2−12−1=3(2−1)2−1\large\frac{3}{\sqrt{2}+1}=\frac{3}{\sqrt{2}+1}\times \frac{\sqrt{2}-1}{\sqrt{2}-1}=\frac{3(\sqrt{2}-1)}{2-1}=32−31\large\hspace8em = \frac{3\sqrt{2}-3}{1}=32−3\large\hspace8em=\normalsize 3\sqrt{2}-3Contoh 2:65−3=65−3×5+35+3\large\frac{6}{\sqrt{5}-\sqrt{3}}=\frac{6}{\sqrt{5}-\sqrt{3}}\times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}=6(5+3)5−3\large\hspace2.4em =\frac{6(\sqrt{5}+\sqrt{3})}{5-3}=6(5+3)2=63(5+3)2\large\hspace2.4em =\frac{6(\sqrt{5}+\sqrt{3})}{2}=\frac{\bcancel{6}^3(\sqrt{5}+\sqrt{3})}{\bcancel2}=35+33\large\hspace2.4em =\normalsize 3\sqrt{5}+3\sqrt{3}
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