$$ \color{blue} a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

Bentuk eksponen \(a^{\frac{1}{2}} = \sqrt{a}\)

Contoh:

\(2^{\frac{1}{2}} = \sqrt[2]{2}=\sqrt{2}\)

\(3^{\frac{5}{7}} = \sqrt[7]{3^5}\)

\(5^{\frac{1}{3}} = \sqrt[3]{5}\)

\(7^{\frac{2}{3}} = \sqrt[3]{7^2}\)

Penjumlahan dan Pengurangan Bentuk Akar

Hanya bentuk akar yang sejenis (angka di dalam akar sama) yang koefisiennya bisa saling dijumlahkan atau dikurangkan untuk disederhanakan.

\(a\sqrt{c} + b\sqrt{c} = (a + b)\sqrt{c}\)

\(a\sqrt{c} \:-\: b\sqrt{c} = (a \:-\: b)\sqrt{c}\)

Contoh:

\(3\sqrt{5} + 4\sqrt{5} = (3 + 4)\sqrt{5} = 7\sqrt{5}\)

\(12\sqrt{7} \:-\: 2\sqrt{7} = (12\:-\:2)\sqrt{7} = 10\sqrt{7}\)

Perkalian Bentuk Akar

\(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)

\(\sqrt{a} \times \sqrt{a} = \sqrt{a^2} = a\)

Contoh:

\(\sqrt{5} \times \sqrt{11} = \sqrt{5 \times 11} = \sqrt{11}\)

\(\sqrt{7} \times \sqrt{7} = \sqrt{7^2} = 7\)

\(5\sqrt{2} \times 7\sqrt{3} = (5\times 7)\sqrt{2 \times 3} = 35\sqrt{6}\)

Pembagian Bentuk Akar

\(\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}\)

Dengan \(a \geq 0 \text{ dan } b \geq 0\)

Contoh:

\(\dfrac{\sqrt{16}}{\sqrt{8}} = \sqrt{\dfrac{16}{8}} = \sqrt{2}\)

\(\dfrac{10\sqrt{8}}{5\sqrt{2}} = \dfrac{10}{5} \cdot \sqrt{\dfrac{8}{2}} = 2\sqrt{4} = 2 \times 2 = 4\)

Merasionalkan Penyebut

(1)  Bentuk \(\dfrac{a}{\sqrt{b}}\)

Kalikan pembilang dan penyebut dengan \(\color{blue}\sqrt{b}\)

\(\dfrac{a}{\sqrt{b}}\times \color{blue}\dfrac{\sqrt{b}}{\sqrt{b}}\)

Contoh:

Rasionalkan \(\dfrac{3}{\sqrt{5}}\)

\(\dfrac{3}{\sqrt{5}}\times \color{blue}\dfrac{\sqrt{5}}{\sqrt{5}} \color{black} = \dfrac{3\sqrt{5}}{5}\)

(2)  Bentuk \(\dfrac{a}{b + \sqrt{c}}\)

Kalikan pembilang dan penyebut dengan \(\color{blue}b\:-\:\sqrt{c}\)

\(\dfrac{a}{b + \sqrt{c}} \times \color{blue} \dfrac{b\:-\:\sqrt{c}}{b\:-\:\sqrt{c}}\)

\(\dfrac{a(b\:-\:\sqrt{c})}{b^2\:-\: (\sqrt{c})^2}\)

\(\dfrac{a(b\:-\:\sqrt{c})}{b^2 \:-\: c}\)

Contoh:

Rasionalkan \(\dfrac{2}{3 + \sqrt{5}}\)

\(\dfrac{2}{3 + \sqrt{5}} \times \color{blue} \dfrac{3 \:-\: \sqrt{5}}{3 \:-\: \sqrt{5}}\)

\(\dfrac{2(3 \:-\: \sqrt{5})}{(3 + \sqrt{5})(3 \:-\: \sqrt{5})}\)

\(\dfrac{2(3 \:-\: \sqrt{5})}{3^2 \:-\:(\sqrt{5})^2}\)

\(\dfrac{6\:-\:2\sqrt{5}}{9 \:-\:5}\)

\(\dfrac{6\:-\:2\sqrt{5}}{4} = \dfrac{6}{4}\:-\:\dfrac{2\sqrt{5}}{4}\)

\(\dfrac{3}{2}\:-\:\dfrac{\sqrt{5}}{2}\)

(3)  Bentuk \(\dfrac{a}{b \:-\: \sqrt{c}}\)

Kalikan pembilang dan penyebut dengan \(\color{blue}b + \sqrt{c}\)

\(\dfrac{a}{b \:-\:\sqrt{c}} \times \color{blue} \dfrac{b + \sqrt{c}}{b + \sqrt{c}}\)

\(\dfrac{a(b + \sqrt{c})}{b^2\:-\: (\sqrt{c})^2}\)

\(\dfrac{a(b + \sqrt{c})}{b^2 \:-\: c}\)

Contoh:

Rasionalkan \(\dfrac{2}{3 \:-\: \sqrt{5}}\)

\(\dfrac{2}{3 \:-\: \sqrt{5}} \times \color{blue} \dfrac{3 +\sqrt{5}}{3 +\sqrt{5}}\)

\(\dfrac{2(3 + \sqrt{5})}{(3 \:-\: \sqrt{5})(3+ \sqrt{5})}\)

\(\dfrac{2(3 + \sqrt{5})}{3^2 \:-\:(\sqrt{5})^2}\)

\(\dfrac{6+2\sqrt{5}}{9 \:-\:5}\)

\(\dfrac{6+2\sqrt{5}}{4} = \dfrac{6}{4}+\dfrac{2\sqrt{5}}{4}\)

\(\dfrac{3}{2}+\dfrac{\sqrt{5}}{2}\)