# Nakafa Framework: LLM

URL: https://nakafa.com/en/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/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Rationalizing Radicals",
  description: "Eliminate radicals from denominators using conjugate multiplication. Master techniques for monomial and binomial cases with comprehensive examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/04/2025",
  subject: "Exponential and Logarithm",
};

## Basic Concepts

Rationalizing radical forms is the process of transforming mathematical expressions that have radicals in the denominator into simpler forms without radicals in the denominator. This is done by multiplying both the numerator and denominator by an appropriate form.

- Rationalizing the Form <InlineMath math="\frac{a}{\sqrt{b}}" />

  To rationalize the form <InlineMath math="\frac{a}{\sqrt{b}}" />, we multiply by its conjugate <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}" />

  Example:

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

- Rationalizing the Form <InlineMath math="\frac{c}{\sqrt{a} + \sqrt{b}}" />

  To rationalize the form <InlineMath math="\frac{c}{\sqrt{a} + \sqrt{b}}" />, we multiply by its conjugate <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}" />

  Example:

  <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}" />

- Rationalizing the Form <InlineMath math="\frac{c}{\sqrt{a} - \sqrt{b}}" />

  To rationalize the form <InlineMath math="\frac{c}{\sqrt{a} - \sqrt{b}}" />, we multiply by its conjugate <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}" />

  Example:

  <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}" />

- Rationalizing the Form <InlineMath math="\frac{c}{a + \sqrt{b}}" />

  To rationalize the form <InlineMath math="\frac{c}{a + \sqrt{b}}" />, we multiply by its conjugate <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}" />

  Example:

  <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}" />

- Rationalizing the Form <InlineMath math="\frac{c}{a - \sqrt{b}}" />

  To rationalize the form <InlineMath math="\frac{c}{a - \sqrt{b}}" />, we multiply by its conjugate <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}" />

  Example:

  <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})" />

## Simplification Exercises

1. Simplify the following expression:

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

2. Simplify the following expression:

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

3. Simplify the following expression:

   <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}}" />

### Answer Key for Simplification

1. Answer:

   <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. Answer:

   <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. Answer:

   <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>

## Rationalization Exercises

1. Rationalize the following expression:

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

2. Rationalize the following expression:

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

3. Rationalize the following expression:

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

### Answer Key for Rationalization

1. Answer:

   <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. Answer:

   <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. Answer:

   <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>

## Benefits of Rationalizing Radical Forms

Rationalizing radical forms has several benefits:

1. Simplifying mathematical expressions
2. Facilitating approximation calculations
3. Eliminating radicals in the denominator to avoid errors in calculations