Merasionalkan bentuk akar adalah menghilangkan bentuk akar pada penyebut dari sebuah pecahan.
Bila penyebut berupa akar tunggal, maka pecahan dikalikan dengan dengan bentuk akar yang dapat menghilangkan akar tersebut.
Contoh 01
Rasionalkan bentuk \( \dfrac{1}{\sqrt{5}}\)
\begin{equation*}
\begin{split}
& \frac{1}{\sqrt{5}} \quad {\color {blue} \times \frac{\sqrt{5}}{\sqrt{5}}} \\\\
& \frac{\sqrt{5}}{5} \\\\
& \frac{1}{5} \sqrt{5}
\end{split}
\end{equation*}
Contoh 02
Rasionalkan bentuk \( \dfrac{1}{\sqrt [3] {7}}\)
\begin{equation*}
\begin{split}
& \frac{1}{\sqrt [3] {7}} \quad {\color {blue} \times \frac{\sqrt [3] {7^2}}{\sqrt [3] {7^2}}} \\\\
& \frac{\sqrt [3] {7^2}}{\sqrt [3] {7^3}} \\\\
& \frac{1}{7} \sqrt [3] {49}
\end{split}
\end{equation*}
Contoh 03
Rasionalkan bentuk \( \dfrac{1}{\sqrt [5] {8}}\)
\begin{equation*}
\begin{split}
& \frac{1}{\sqrt [5] {8}} \\\\
& \frac{1}{\sqrt [5] {2^3}} \quad {\color {blue} \times \frac{\sqrt [5] {2^2}}{\sqrt [5] {2^2}}} \\\\
& \frac{\sqrt [5] {2^2}}{\sqrt [5] {2^5}} \\\\
& \frac{1}{2} \sqrt [5] {4}
\end{split}
\end{equation*}
Bila penyebut berupa penjumlahan akar, maka pecahan dikalikan dengan dengan sekawannya.
| PENYEBUT | SEKAWAN |
| \(\sqrt {a} + \sqrt{b}\) | \(\sqrt {a} - \sqrt{b}\) |
| \(\sqrt {a} - \sqrt{b}\) | \(\sqrt {a} + \sqrt{b}\) |
| \(\sqrt [3] {a} + \sqrt [3] {b}\) | \(\sqrt [3] {a^2} - \sqrt [3] {ab} + \sqrt [3] {b^2}\) |
| \(\sqrt [3] {a} - \sqrt [3] {b}\) | \(\sqrt [3] {a^2} + \sqrt [3] {ab} + \sqrt [3] {b^2}\) |
Contoh 04
\begin{equation*}
\begin{split}
& \frac{1}{\sqrt{5} + \sqrt{2}} \quad {\color {blue} \times \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}} \\\\
& \frac{\sqrt{5} - \sqrt{2}}{(\sqrt{5})^2 - (\sqrt{2})^2} \\\\
& \frac{\sqrt{5} - \sqrt{2}}{5 - 2} \\\\
& \frac{\sqrt{5} - \sqrt{2}}{3} \\\\
& \frac{1}{3} (\sqrt{5} - \sqrt{2})
\end{split}
\end{equation*}
Contoh 05
\begin{equation*}
\begin{split}
& \frac{1}{\sqrt [3] {7} - \sqrt [3] {2}} \\\\
& \frac{1}{\sqrt [3] {7} - \sqrt [3] {2}} \quad {\color {blue} \times \frac{\sqrt [3] {7^2} + \sqrt [3] {7 \:.\: 2} + \sqrt [3] {2^2}}{\sqrt [3] {7^2} + \sqrt [3] {7 \:.\: 2} + \sqrt [3] {2^2}}} \\\\
& \frac{\sqrt [3] {7^2} + \sqrt [3] {7 \:.\: 2} + \sqrt [3] {2^2}}{7 + 3} \\\\
& \frac {1}{10} (\sqrt [3] {49} + \sqrt [3] {21} + \sqrt [3] {4})
\end{split}
\end{equation*}
| RUMUS YANG DIGUNAKAN | |
| \((a + b)(a - b) = a^2 - b^2\) | \((\sqrt {a} + \sqrt{b})(\sqrt {a} - \sqrt{b}) = a - b\) |
| \((a + b)(a^2 - ab + b^2) = a^3 + b^3\) | \((\sqrt [3] {a} + \sqrt [3] {b})(\sqrt [3] {a^2} - \sqrt [3] {ab} + \sqrt [3] {b^2}) = a + b\) |
| \((a - b)(a^2 + ab + b^2) = a^3 - b^3\) | \((\sqrt [3] {a} - \sqrt [3] {b})(\sqrt [3] {a^2} + \sqrt [3] {ab} + \sqrt [3] {b^2}) = a - b\) |