A. BENTUK EKSPONEN
Bentuk eksponen dapat disederhanakan dengan menggunakan aturan eksponen di bawah ini:
| Aturan | Contoh |
|---|---|
| \(\large x^a \:.\: x^b = x^{a + b}\) |
\(\large x^5 \:.\: x^2 = x^{5 + 2} = x^7\)
|
| \(\large \dfrac{x^a}{x^b} = x^{a - b}\) |
\(\large \dfrac {x^5}{x^2} = x^{5 - 2} = x^3\)
|
| \(\large (x^a)^b = x^{a \:.\: b}\) |
\(\large \left(x^5\right)^2 = x^{5 \:.\: 2} = x^{10}\)
|
| \(\large x^a \:.\: y^a = (x \:.\: y)^a\) |
\(\large x^3 \:.\: y^3 = (x \:.\: y)^3\)
|
| \(\large \dfrac {x^a}{y^a} = \left(\dfrac xy \right)^a \) |
\(\large \dfrac {x^3}{y^3} = \left(\dfrac xy \right)^3 \)
|
| \(\large x^{-a} = \dfrac {1}{x^a}\) |
\(\large x^{-5} = \dfrac {1}{x^5}\)
|
| \(\large \left (\dfrac xy \right)^{-a} = \left (\dfrac yx \right)^a\) |
\(\large \left (\dfrac xy \right)^{-3} = \left (\dfrac yx \right)^3\)
|
| \(\large x^{\frac ab} = \sqrt [b] {x^a}\) |
\(\large x^{\frac 12} = \sqrt {x}\)
\(\large x^{\frac 25} = \sqrt [5] {x^2}\)
|
B. BENTUK AKAR
1. Menyederhanakan Bentuk Akar
Bentuk akar disederhanakan dengan memisahkan bilangan yang dapat diakar.
\(\sqrt {12} = \sqrt {4 \:.\: 3} = 2 \sqrt{3}\)
\(\sqrt {50} = \sqrt {25 \:.\: 2} = 5 \sqrt{2}\)
\(\sqrt [3] {54} = \sqrt [3] {27 \:.\: 2} = 3 \sqrt [3] {2}\)
2. 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} \\\\ & \bbox[5px, border: 2px solid magenta] {\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}} \\\\ & \bbox[5px, border: 2px solid magenta] {\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}} \\\\ & \bbox[5px, border: 2px solid magenta] {\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} \\\\ & \bbox[5px, border: 2px solid magenta] {\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} \\\\ & \bbox[5px, border: 2px solid magenta] {\frac {1}{10} (\sqrt [3] {49} + \sqrt [3] {21} + \sqrt [3] {4})} \end{split} \end{equation*}
| RUMUS YANG DIGUNAKAN |
|---|
| \((a + b)(a - b) = a^2 - b^2\) |
| \((a + b)(a^2 - ab + b^2) = a^3 + b^3\) |
| \((a - b)(a^2 + ab + b^2) = a^3 - b^3\) |
3. 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
\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
\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*}
C. PERSAMAAN EKSPONEN SEDERHANA
Pada persamaan bentuk eksponen, kita diminta untuk menentukan nilai variabel tertentu yang memenuhi persamaan tersebut.
Contoh 01
Tentukan nilai \(x\) yang memenuhi dari \(2^x = 32\)
\begin{equation*} \begin{split} & 2^x = 32 \\\\ & 2^x = 2^5 \\\\ & \bbox[5px, border: 2px solid magenta] {x = 5} \end{split} \end{equation*}
Contoh 02
Tentukan nilai \(x\) yang memenuhi dari \(\sqrt [x] {27} = 9\)
\begin{equation*} \begin{split} & \sqrt [x] {27} = 9 \\\\ & \sqrt [x] {3^3} = 3^2 \\\\ & 3^{\frac 3x} = 3^2 \\\\ & \frac 3x = 2 \\\\ & \bbox[5px, border: 2px solid magenta] {x = \frac 32} \end{split} \end{equation*}
A. Bentuk Eksponen B. Bentuk Akar C. Persamaan Eksponen Sederhana D. Persiapan Ulangan Kembali ke Modul SMA