Merasionalkan Bentuk Akar

Konsep Dasar

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
\(\sqrt {a} + \sqrt{b}\) \(\sqrt {a} - \sqrt{b}\)
\(\sqrt {a} - \sqrt{b}\) \(\sqrt {a} + \sqrt{b}\)
\(\sqrt [3] {a} + \sqrt [3] {b}\) \(\sqrt [3] {a^2} - \sqrt [3] {ab} + \sqrt [3] {b^2}\)
\(\sqrt [3] {a} - \sqrt [3] {b}\) \(\sqrt [3] {a^2} + \sqrt [3] {ab} + \sqrt [3] {b^2}\)

 

RUMUS YANG DIGUNAKAN
\((a + b)(a - b) = a^2 - b^2\) \((\sqrt {a} + \sqrt{b})(\sqrt {a} - \sqrt{b}) = a - b\)
\((a + b)(a^2 - ab + b^2) = a^3 + b^3\) \((\sqrt [3] {a} + \sqrt [3] {b})(\sqrt [3] {a^2} - \sqrt [3] {ab} + \sqrt [3] {b^2}) = a + b\)
\((a - b)(a^2 + ab + b^2) = a^3 - b^3\) \((\sqrt [3] {a} - \sqrt [3] {b})(\sqrt [3] {a^2} + \sqrt [3] {ab} + \sqrt [3] {b^2}) = a - b\)

 

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*}

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