# Nakafa Framework: LLM

URL: /id/subject/high-school/10/mathematics/exponential-logarithm/rationalizing-radicals
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/exponential-logarithm/rationalizing-radicals/id.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Merasionalkan Akar",
  description: "Hilangkan akar dari penyebut menggunakan perkalian sekawan. Kuasai teknik untuk kasus monomial dan binomial dengan contoh menyeluruh.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/04/2025",
  subject: "Eksponen dan Logaritma",
};

## Konsep Dasar

Merasionalkan bentuk akar adalah proses mengubah ekspresi matematika yang memiliki akar di penyebut menjadi bentuk yang lebih sederhana tanpa akar di penyebut. Hal ini dilakukan dengan mengalikan pembilang dan penyebut dengan bentuk yang sesuai.

- Merasionalkan Bentuk <InlineMath math="\frac{a}{\sqrt{b}}" />

  Untuk merasionalkan bentuk <InlineMath math="\frac{a}{\sqrt{b}}" />, kita kalikan dengan sekawannya yaitu <InlineMath math="\frac{\sqrt{b}}{\sqrt{b}}" />:

  <BlockMath math="\frac{a}{\sqrt{b}} = \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{\sqrt{b} \times \sqrt{b}} = \frac{a\sqrt{b}}{b}" />

  Contoh:

  <BlockMath math="\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}" />

- Merasionalkan Bentuk <InlineMath math="\frac{c}{\sqrt{a} + \sqrt{b}}" />

  Untuk merasionalkan bentuk <InlineMath math="\frac{c}{\sqrt{a} + \sqrt{b}}" />, kita kalikan dengan sekawannya <InlineMath math="\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}" />:

  <BlockMath math="\frac{c}{\sqrt{a} + \sqrt{b}} = \frac{c}{\sqrt{a} + \sqrt{b}} \times \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}} = \frac{c(\sqrt{a} - \sqrt{b})}{(\sqrt{a})^2 - (\sqrt{b})^2} = \frac{c(\sqrt{a} - \sqrt{b})}{a - b}" />

  Contoh:

  <BlockMath math="\frac{4}{\sqrt{7} + \sqrt{3}} = \frac{4}{\sqrt{7} + \sqrt{3}} \times \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}} = \frac{4(\sqrt{7} - \sqrt{3})}{7 - 3} = \frac{4(\sqrt{7} - \sqrt{3})}{4} = \sqrt{7} - \sqrt{3}" />

- Merasionalkan Bentuk <InlineMath math="\frac{c}{\sqrt{a} - \sqrt{b}}" />

  Untuk merasionalkan bentuk <InlineMath math="\frac{c}{\sqrt{a} - \sqrt{b}}" />, kita kalikan dengan sekawannya <InlineMath math="\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}}" />:

  <BlockMath math="\frac{c}{\sqrt{a} - \sqrt{b}} = \frac{c}{\sqrt{a} - \sqrt{b}} \times \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}} = \frac{c(\sqrt{a} + \sqrt{b})}{(\sqrt{a})^2 - (\sqrt{b})^2} = \frac{c(\sqrt{a} + \sqrt{b})}{a - b}" />

  Contoh:

  <BlockMath math="\frac{2}{\sqrt{5} - \sqrt{2}} = \frac{2}{\sqrt{5} - \sqrt{2}} \times \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}} = \frac{2(\sqrt{5} + \sqrt{2})}{5 - 2} = \frac{2(\sqrt{5} + \sqrt{2})}{3}" />

- Merasionalkan Bentuk <InlineMath math="\frac{c}{a + \sqrt{b}}" />

  Untuk merasionalkan bentuk <InlineMath math="\frac{c}{a + \sqrt{b}}" />, kita kalikan dengan sekawannya <InlineMath math="\frac{a - \sqrt{b}}{a - \sqrt{b}}" />:

  <BlockMath math="\frac{c}{a + \sqrt{b}} = \frac{c}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{c(a - \sqrt{b})}{a^2 - b} = \frac{c(a - \sqrt{b})}{a^2 - b}" />

  Contoh:

  <BlockMath math="\frac{6}{4 + \sqrt{3}} = \frac{6}{4 + \sqrt{3}} \times \frac{4 - \sqrt{3}}{4 - \sqrt{3}} = \frac{6(4 - \sqrt{3})}{16 - 3} = \frac{6(4 - \sqrt{3})}{13}" />

- Merasionalkan Bentuk <InlineMath math="\frac{c}{a - \sqrt{b}}" />

  Untuk merasionalkan bentuk <InlineMath math="\frac{c}{a - \sqrt{b}}" />, kita kalikan dengan sekawannya <InlineMath math="\frac{a + \sqrt{b}}{a + \sqrt{b}}" />:

  <BlockMath math="\frac{c}{a - \sqrt{b}} = \frac{c}{a - \sqrt{b}} \times \frac{a + \sqrt{b}}{a + \sqrt{b}} = \frac{c(a + \sqrt{b})}{a^2 - b} = \frac{c(a + \sqrt{b})}{a^2 - b}" />

  Contoh:

  <BlockMath math="\frac{3}{2 - \sqrt{5}} = \frac{3}{2 - \sqrt{5}} \times \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{3(2 + \sqrt{5})}{4 - 5} = \frac{3(2 + \sqrt{5})}{-1} = -3(2 + \sqrt{5})" />

## Latihan Menyederhanakan

1. Sederhanakan bentuk berikut:

   <BlockMath math="\left(\frac{8x^5y^{-4}}{16y^{-\frac{1}{4}}}\right)^{\frac{1}{2}}" />

2. Sederhanakan bentuk berikut:

   <BlockMath math="\left(5\sqrt{x^5}\right)\left(3\sqrt[3]{x}\right)" />

3. Sederhanakan bentuk berikut:

   <BlockMath math="\left(\frac{p^5q^{-10}}{p^5q^{-4}}\right)^{\frac{1}{2}}\left(\frac{p^{\frac{1}{4}}q^{-\frac{1}{2}}}{p^{-\frac{1}{2}}q^{-\frac{1}{2}}}\right)^{\frac{1}{2}}" />

### Kunci Jawaban Menyederhanakan

1. Jawaban:

   <div className="flex flex-col gap-4">
     <BlockMath math="\left(\frac{8x^5y^{-4}}{16y^{-\frac{1}{4}}}\right)^{\frac{1}{2}} = \frac{\left(2^3\right)^{\frac{1}{2}}\left(x^5\right)^{\frac{1}{2}}\left(y^{-4}\right)^{\frac{1}{2}}}{\left(2^4\right)^{\frac{1}{2}}\left(y^{-\frac{1}{4}}\right)^{\frac{1}{2}}}" />
     <BlockMath math="= \frac{\left(2\right)^{\frac{3}{2}}\left(x\right)^{\frac{5}{2}}\left(y\right)^{-2}}{2^2\left(y\right)^{-\frac{1}{8}}}" />
     <BlockMath math="= \left(2\right)^{\frac{3}{2}-2}\left(x\right)^{\frac{5}{2}}\left(y\right)^{-2+\frac{1}{8}}" />
     <BlockMath math="= \frac{\left(x\right)^{\frac{5}{2}}}{\left(2\right)^{\frac{1}{2}}\left(y\right)^{\frac{15}{8}}}" />
   </div>

2. Jawaban:

   <div className="flex flex-col gap-4">
     <BlockMath math="\left(5\sqrt{x^5}\right)\left(3\sqrt[3]{x}\right) = \left(5x^{\frac{5}{2}}\right)\left(3x^{\frac{1}{3}}\right)" />
     <BlockMath math="= 15x^{\frac{5}{2}+\frac{1}{3}}" />
     <BlockMath math="= 15x^{\frac{15+2}{6}}" />
     <BlockMath math="= 15x^{\frac{17}{6}}" />
     <BlockMath math="= 15x^{2\frac{5}{6}}" />
     <BlockMath math="= 15x^2\sqrt[6]{x^5}" />
   </div>

3. Jawaban:

   <div className="flex flex-col gap-4">
     <BlockMath math="\left(\frac{p^5q^{-10}}{p^5q^{-4}}\right)^{\frac{1}{2}}\left(\frac{p^{\frac{1}{4}}q^{-\frac{1}{2}}}{p^{-\frac{1}{2}}q^{-\frac{1}{2}}}\right)^{\frac{1}{2}} = \left(\frac{p^{5-5}q^{-10-(-4)}}{1}\right)^{\frac{1}{2}}\left(\frac{p^{\frac{1}{4}-(-\frac{1}{2})}q^{-\frac{1}{2}-(-\frac{1}{2})}}{1}\right)^{\frac{1}{2}}" />
     <BlockMath math="= \left(p^0q^{-6}\right)^{\frac{1}{2}}\left(p^{\frac{3}{4}}q^0\right)^{\frac{1}{2}}" />
     <BlockMath math="= \left(p^0q^{-3}\right)\left(p^{\frac{3}{8}}\cdot 1\right)" />
     <BlockMath math="= \left(1 \cdot q^{-3}\right)\left(p^{\frac{3}{8}} \cdot 1\right)" />
     <BlockMath math="= \frac{p^{\frac{3}{8}}}{q^3}" />
     <BlockMath math="= \frac{\sqrt[8]{p^3}}{q^3}" />
   </div>

## Latihan Merasionalkan

1. Merasionalkan bentuk berikut:

   <BlockMath math="\frac{2}{\sqrt{b^3}}" />

2. Merasionalkan bentuk berikut:

   <BlockMath math="\frac{2}{\sqrt{3} + \sqrt{5}}" />

3. Merasionalkan bentuk berikut:

   <BlockMath math="\frac{m}{\sqrt{m} + n}" />

### Kunci Jawaban Merasionalkan

1. Jawaban:

   <div className="flex flex-col gap-4">
     <BlockMath math="\frac{2}{\sqrt{b^3}} = \frac{2}{\sqrt{b^3}} \times \frac{\sqrt{b}}{\sqrt{b}}" />
     <BlockMath math="= \frac{2\sqrt{b}}{\sqrt{b^4}}" />
     <BlockMath math="= \frac{2\sqrt{b}}{b^2}" />
   </div>

2. Jawaban:

   <div className="flex flex-col gap-4">
     <BlockMath math="\frac{2}{\sqrt{3} + \sqrt{5}} = \frac{2}{\sqrt{3} + \sqrt{5}} \times \frac{\sqrt{3} - \sqrt{5}}{\sqrt{3} - \sqrt{5}}" />
     <BlockMath math="= \frac{2(\sqrt{3} - \sqrt{5})}{(\sqrt{3})^2 - (\sqrt{5})^2}" />
     <BlockMath math="= \frac{2(\sqrt{3} - \sqrt{5})}{3 - 5}" />
     <BlockMath math="= \frac{2(\sqrt{3} - \sqrt{5})}{-2}" />
     <BlockMath math="= -(\sqrt{3} - \sqrt{5})" />
     <BlockMath math="= \sqrt{5} - \sqrt{3}" />
   </div>

3. Jawaban:

   <div className="flex flex-col gap-4">
     <BlockMath math="\frac{m}{\sqrt{m} + n} = \frac{m}{\sqrt{m} + n} \times \frac{\sqrt{m} - n}{\sqrt{m} - n}" />
     <BlockMath math="= \frac{m(\sqrt{m} - n)}{(\sqrt{m})^2 - n^2}" />
     <BlockMath math="= \frac{m(\sqrt{m} - n)}{m - n^2}" />
   </div>

## Manfaat Merasionalkan Bentuk Akar

Merasionalkan bentuk akar memiliki beberapa manfaat:

1. Menyederhanakan bentuk ekspresi matematika
2. Memudahkan perhitungan nilai pendekatan
3. Menghilangkan bentuk akar di penyebut untuk menghindari kesalahan dalam perhitungan
