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Merasionalkan Penyebut Bentuk Akar

Tujuan Pembelajaran

Setelah mempelajari materi ini, siswa diharapkan mampu:

  1. Menjelaskan konsep merasionalkan penyebut bentuk akar
  2. Menentukan bentuk sekawan dari suatu bentuk akar
  3. Merasionalkan penyebut $\frac{a}{\sqrt{b}}$
  4. Merasionalkan penyebut $\frac{a}{b \pm \sqrt{c}}$
  5. Merasionalkan penyebut $\frac{a}{\sqrt{b} \pm \sqrt{c}}$

A. Konsep Dasar

Merasionalkan penyebut berarti mengubah penyebut yang berbentuk akar menjadi bilangan rasional.

Dasar dari merasionalkan penyebut adalah perkalian dengan bentuk sekawan:

$$a\sqrt{b} \times \sqrt{b} = a \times b$$

Bentuk Sekawan

BentukSekawannyaHasil Perkalian
$\sqrt{b}$$\sqrt{b}$$b$ (rasional)
$b + \sqrt{c}$$b - \sqrt{c}$$b^2 - c$ (rasional)
$b - \sqrt{c}$$b + \sqrt{c}$$b^2 - c$ (rasional)
$\sqrt{b} + \sqrt{c}$$\sqrt{b} - \sqrt{c}$$b - c$ (rasional)
$\sqrt{b} - \sqrt{c}$$\sqrt{b} + \sqrt{c}$$b - c$ (rasional)

Aturan: Kalikan pembilang dan penyebut dengan bentuk sekawan penyebut. Jangan lupa kalikan juga pembilangnya!

B. Tipe 1: Penyebut $\sqrt{b}$

Rumus

$$\frac{a}{\sqrt{b}} = \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$$

Contoh 1

SoalProsesHasil
$\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$$\frac{\sqrt{2}}{2}$
$\frac{3}{\sqrt{5}}$$\frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$$\frac{3\sqrt{5}}{5}$
$\frac{7}{\sqrt{3}}$$\frac{7}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$$\frac{7\sqrt{3}}{3}$
$\frac{2}{\sqrt{6}}$$\frac{2}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}$$\frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3}$
$\frac{5}{\sqrt{10}}$$\frac{5}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}}$$\frac{5\sqrt{10}}{10} = \frac{\sqrt{10}}{2}$

Contoh 2: Dengan Variabel

SoalProsesHasil
$\frac{2x}{\sqrt{3}}$$\frac{2x}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$$\frac{2x\sqrt{3}}{3}$
$\frac{a}{\sqrt{a}}$$\frac{a}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}}$$\frac{a\sqrt{a}}{a} = \sqrt{a}$

C. Tipe 2: Penyebut $b \pm \sqrt{c}$

Rumus

$$\frac{a}{b + \sqrt{c}} = \frac{a}{b + \sqrt{c}} \times \frac{b - \sqrt{c}}{b - \sqrt{c}} = \frac{a(b - \sqrt{c})}{b^2 - c}$$

$$\frac{a}{b - \sqrt{c}} = \frac{a}{b - \sqrt{c}} \times \frac{b + \sqrt{c}}{b + \sqrt{c}} = \frac{a(b + \sqrt{c})}{b^2 - c}$$

Contoh 3

SoalProsesHasil
$\frac{2}{1 + \sqrt{3}}$$\frac{2}{1 + \sqrt{3}} \times \frac{1 - \sqrt{3}}{1 - \sqrt{3}}$$\frac{2(1 - \sqrt{3})}{1 - 3} = \frac{2 - 2\sqrt{3}}{-2} = \sqrt{3} - 1$
$\frac{5}{3 - \sqrt{2}}$$\frac{5}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}}$$\frac{5(3 + \sqrt{2})}{9 - 2} = \frac{15 + 5\sqrt{2}}{7}$
$\frac{4}{\sqrt{5} + 2}$$\frac{4}{\sqrt{5} + 2} \times \frac{\sqrt{5} - 2}{\sqrt{5} - 2}$$\frac{4(\sqrt{5} - 2)}{5 - 4} = 4\sqrt{5} - 8$
$\frac{6}{\sqrt{7} - \sqrt{3}}$—(lihat Tipe 3)

D. Tipe 3: Penyebut $\sqrt{b} \pm \sqrt{c}$

Rumus

$$\frac{a}{\sqrt{b} + \sqrt{c}} = \frac{a}{\sqrt{b} + \sqrt{c}} \times \frac{\sqrt{b} - \sqrt{c}}{\sqrt{b} - \sqrt{c}} = \frac{a(\sqrt{b} - \sqrt{c})}{b - c}$$

$$\frac{a}{\sqrt{b} - \sqrt{c}} = \frac{a}{\sqrt{b} - \sqrt{c}} \times \frac{\sqrt{b} + \sqrt{c}}{\sqrt{b} + \sqrt{c}} = \frac{a(\sqrt{b} + \sqrt{c})}{b - c}$$

Contoh 4

SoalProsesHasil
$\frac{3}{\sqrt{5} - \sqrt{2}}$$\frac{3}{\sqrt{5} - \sqrt{2}} \times \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}}$$\frac{3(\sqrt{5} + \sqrt{2})}{5 - 2} = \frac{3\sqrt{5} + 3\sqrt{2}}{3} = \sqrt{5} + \sqrt{2}$
$\frac{6}{\sqrt{7} - \sqrt{3}}$$\frac{6}{\sqrt{7} - \sqrt{3}} \times \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}}$$\frac{6(\sqrt{7} + \sqrt{3})}{7 - 3} = \frac{6\sqrt{7} + 6\sqrt{3}}{4} = \frac{3\sqrt{7} + 3\sqrt{3}}{2}$
$\frac{8}{\sqrt{10} + \sqrt{2}}$$\frac{8}{\sqrt{10} + \sqrt{2}} \times \frac{\sqrt{10} - \sqrt{2}}{\sqrt{10} - \sqrt{2}}$$\frac{8(\sqrt{10} - \sqrt{2})}{10 - 2} = \frac{8\sqrt{10} - 8\sqrt{2}}{8} = \sqrt{10} - \sqrt{2}$

E. Rangkuman

NoTipe PenyebutSekawanHasil
1$\sqrt{b}$$\sqrt{b}$$\frac{a\sqrt{b}}{b}$
2$b + \sqrt{c}$$b - \sqrt{c}$$\frac{a(b - \sqrt{c})}{b^2 - c}$
3$b - \sqrt{c}$$b + \sqrt{c}$$\frac{a(b + \sqrt{c})}{b^2 - c}$
4$\sqrt{b} + \sqrt{c}$$\sqrt{b} - \sqrt{c}$$\frac{a(\sqrt{b} - \sqrt{c})}{b - c}$
5$\sqrt{b} - \sqrt{c}$$\sqrt{b} + \sqrt{c}$$\frac{a(\sqrt{b} + \sqrt{c})}{b - c}$
💡 Tips: Ingat rumus sekawan: $(p+q)(p-q) = p^2 - q^2$. Ini kunci merasionalkan semua jenis penyebut! Pastikan setelah merasionalkan, kamu **menyederhanakan** hasilnya jika masih bisa.

F. Latihan Soal

Rasionalkan penyebut dari bentuk berikut!

  1. $\frac{2}{\sqrt{3}}$
  2. $\frac{5}{\sqrt{5}}$
  3. $\frac{4}{\sqrt{8}}$
  4. $\frac{1}{2 + \sqrt{3}}$
  5. $\frac{3}{4 - \sqrt{5}}$
  6. $\frac{7}{\sqrt{7} + \sqrt{2}}$
  7. $\frac{10}{\sqrt{11} - \sqrt{6}}$
  8. $\frac{2\sqrt{3}}{\sqrt{6}}$
  9. $\frac{\sqrt{2}}{3 - \sqrt{2}}$
  10. $\frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{3} - \sqrt{2}}$

G. Kunci Jawaban

  1. $\frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$
  2. $\frac{5}{\sqrt{5}} = \sqrt{5}$
  3. $\frac{4}{\sqrt{8}} = \frac{4}{2\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$ atau $\frac{4\sqrt{8}}{8} = \frac{4 \times 2\sqrt{2}}{8} = \sqrt{2}$
  4. $\frac{1}{2 + \sqrt{3}} = 2 - \sqrt{3}$
  5. $\frac{3}{4 - \sqrt{5}} = \frac{3(4 + \sqrt{5})}{16 - 5} = \frac{12 + 3\sqrt{5}}{11}$
  6. $\frac{7}{\sqrt{7} + \sqrt{2}} = \frac{7(\sqrt{7} - \sqrt{2})}{7 - 2} = \frac{7\sqrt{7} - 7\sqrt{2}}{5}$
  7. $\frac{10}{\sqrt{11} - \sqrt{6}} = \frac{10(\sqrt{11} + \sqrt{6})}{11 - 6} = 2\sqrt{11} + 2\sqrt{6}$
  8. $\frac{2\sqrt{3}}{\sqrt{6}} = 2\sqrt{\frac{3}{6}} = 2\sqrt{\frac{1}{2}} = \sqrt{2}$
  9. $\frac{\sqrt{2}}{3 - \sqrt{2}} = \frac{\sqrt{2}(3 + \sqrt{2})}{9 - 2} = \frac{3\sqrt{2} + 2}{7}$
  10. $\frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} + \frac{\sqrt{3} + \sqrt{2}}{3 - 2} = (\sqrt{3} - \sqrt{2}) + (\sqrt{3} + \sqrt{2}) = 2\sqrt{3}$