Menyederhanakan Bentuk Akar
Bentuk akar disederhanakan dengan memisahkan bilangan yang dapat diakar.
Contoh 01
\begin{equation*} \begin{split} & \sqrt {12} = \sqrt {4 \:.\: 3} \\\\ & \sqrt {12} = 2 \sqrt{3} \end{split} \end{equation*}
Contoh 02
\begin{equation*} \begin{split} & \sqrt {50} = \sqrt {25 \:.\: 2} \\\\ & \sqrt {50} = 5 \sqrt{2} \end{split} \end{equation*}
Contoh 03
\begin{equation*} \begin{split} & \sqrt [3] {54} = \sqrt [3] {27 \:.\: 2} \\\\ & \sqrt [3] {54} = 3 \sqrt [3] {2} \end{split} \end{equation*}
Merasionalkan Bentuk Akar
Merasionalkan bentuk akar adalah menghilangkan bentuk akar pada penyebut dari sebuah pecahan.
Contoh 01
\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
\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
\begin{equation*} \begin{split} & \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 | Hasil Kali |
| \(\sqrt {a} + \sqrt{b}\) | \(\sqrt {a} - \sqrt{b}\) | \(a - b\) |
| \(\sqrt {a} - \sqrt{b}\) | \(\sqrt {a} + \sqrt{b}\) | \(a - b\) |
| \(\sqrt [3] {a} + \sqrt [3] {b}\) | \(\sqrt [3] {a^2} - \sqrt [3] {ab} + \sqrt [3] {b^2}\) | \(a + b\) |
| \(\sqrt [3] {a} - \sqrt [3] {b}\) | \(\sqrt [3] {a^2} + \sqrt [3] {ab} + \sqrt [3] {b^2}\) | \(a - b\) |
Contoh 01
\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}}{5 - 2} \\\\ & \frac{1}{3} (\sqrt{5} - \sqrt{2}) \end{split} \end{equation*}
Contoh 02
\begin{equation*} \begin{split} & \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 | |
| \((a + b)(a - b) = a^2 - b^2\) | \((\sqrt {a} + \sqrt{b})(\sqrt {a} - \sqrt{b}) = a - b\) |
| \((a + b)(a^2 - ab + b^2) = a^3 + b^3\) | \((\sqrt [3] {a} + \sqrt [3] {b})(\sqrt [3] {a^2} - \sqrt [3] {ab} + \sqrt [3] {b^2}) = a + b\) |
| \((a - b)(a^2 + ab + b^2) = a^3 - b^3\) | \((\sqrt [3] {a} - \sqrt [3] {b})(\sqrt [3] {a^2} + \sqrt [3] {ab} + \sqrt [3] {b^2}) = a - b\) |
Bentuk Akar di Dalam Akar
\(\sqrt{(a + b) + 2 \sqrt{ab}} = \sqrt{a} + \sqrt{b}\)
\(\sqrt{(a + b) - 2 \sqrt{ab}} = \sqrt{a} - \sqrt{b}\)
Contoh 01
\begin{equation*} \begin{split} & \sqrt{5 + 2\sqrt{6}} \\\\ & \sqrt{(3 + 2) + 2\sqrt{3 \:.\: 2}} \\\\ & \bbox[5px, border: 2px solid magenta] {\sqrt{3} + \sqrt{2}} \end{split} \end{equation*}
Contoh 02
\begin{equation*} \begin{split} & \sqrt{4 - 2\sqrt{3}} \\\\ & \sqrt{(3 + 1) - 2\sqrt{3 \:.\: 1}} \\\\ & \sqrt{3} - \sqrt{1} \\\\ & \bbox[5px, border: 2px solid magenta] {\sqrt{3} - 1} \end{split} \end{equation*}
Penurunan Rumus
\begin{equation*} \begin{split} (a + b)^2 & = a^2 + 2ab + b^2 \\\\ (\sqrt{a} + \sqrt{b})^2 & = a + 2 \sqrt{a} \sqrt{b} + b \\\\ (\sqrt{a} + \sqrt{b})^2 & = a + b + 2 \sqrt{ab} \\\\ \sqrt{a} + \sqrt{b} & = \sqrt{a + b + 2 \sqrt{ab}} \end{split} \end{equation*}
\begin{equation*} \begin{split} (a - b)^2 & = a^2 - 2ab + b^2 \\\\ (\sqrt{a} - \sqrt{b})^2 & = a - 2 \sqrt{a} \sqrt{b} + b \\\\ (\sqrt{a} - \sqrt{b})^2 & = a + b - 2 \sqrt{ab} \\\\ \sqrt{a} - \sqrt{b} & = \sqrt{a + b - 2 \sqrt{ab}} \end{split} \end{equation*}
SOAL LATIHAN