Rationalizing Radicals

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

  To rationalize the form $\frac{a}{\sqrt{b}}$, we multiply by its conjugate $\frac{\sqrt{b}}{\sqrt{b}}$:

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

  Example:

  $$\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$$

• Rationalizing the Form $\frac{c}{\sqrt{a} + \sqrt{b}}$

  To rationalize the form $\frac{c}{\sqrt{a} + \sqrt{b}}$, we multiply by its conjugate $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$:

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

  Example:

  $$\frac{4}{\sqrt{7} + \sqrt{3}} = \frac{4}{\sqrt{7} + \sqrt{3}} \times \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}} = \frac{4(\sqrt{7} - \sqrt{3})}{7 - 3} = \frac{4(\sqrt{7} - \sqrt{3})}{4} = \sqrt{7} - \sqrt{3}$$

• Rationalizing the Form $\frac{c}{\sqrt{a} - \sqrt{b}}$

  To rationalize the form $\frac{c}{\sqrt{a} - \sqrt{b}}$, we multiply by its conjugate $\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}}$:

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

  Example:

  $$\frac{2}{\sqrt{5} - \sqrt{2}} = \frac{2}{\sqrt{5} - \sqrt{2}} \times \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}} = \frac{2(\sqrt{5} + \sqrt{2})}{5 - 2} = \frac{2(\sqrt{5} + \sqrt{2})}{3}$$

• Rationalizing the Form $\frac{c}{a + \sqrt{b}}$

  To rationalize the form $\frac{c}{a + \sqrt{b}}$, we multiply by its conjugate $\frac{a - \sqrt{b}}{a - \sqrt{b}}$:

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

  Example:

  $$\frac{6}{4 + \sqrt{3}} = \frac{6}{4 + \sqrt{3}} \times \frac{4 - \sqrt{3}}{4 - \sqrt{3}} = \frac{6(4 - \sqrt{3})}{16 - 3} = \frac{6(4 - \sqrt{3})}{13}$$

• Rationalizing the Form $\frac{c}{a - \sqrt{b}}$

  To rationalize the form $\frac{c}{a - \sqrt{b}}$, we multiply by its conjugate $\frac{a + \sqrt{b}}{a + \sqrt{b}}$:

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

  Example:

  $$\frac{3}{2 - \sqrt{5}} = \frac{3}{2 - \sqrt{5}} \times \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{3(2 + \sqrt{5})}{4 - 5} = \frac{3(2 + \sqrt{5})}{-1} = -3(2 + \sqrt{5})$$

Simplification Exercises

1. Simplify the following expression:

   $$\left(\frac{8x^5y^{-4}}{16y^{-\frac{1}{4}}}\right)^{\frac{1}{2}}$$

2. Simplify the following expression:

   $$\left(5\sqrt{x^5}\right)\left(3\sqrt[3]{x}\right)$$

3. Simplify the following expression:

   $$\left(\frac{p^5q^{-10}}{p^5q^{-4}}\right)^{\frac{1}{2}}\left(\frac{p^{\frac{1}{4}}q^{-\frac{1}{2}}}{p^{-\frac{1}{2}}q^{-\frac{1}{2}}}\right)^{\frac{1}{2}}$$

Answer Key for Simplification

1. Answer:

   $$\left(\frac{8x^5y^{-4}}{16y^{-\frac{1}{4}}}\right)^{\frac{1}{2}} = \frac{\left(2^3\right)^{\frac{1}{2}}\left(x^5\right)^{\frac{1}{2}}\left(y^{-4}\right)^{\frac{1}{2}}}{\left(2^4\right)^{\frac{1}{2}}\left(y^{-\frac{1}{4}}\right)^{\frac{1}{2}}}$$

   $$= \frac{\left(2\right)^{\frac{3}{2}}\left(x\right)^{\frac{5}{2}}\left(y\right)^{-2}}{2^2\left(y\right)^{-\frac{1}{8}}}$$

   $$= \left(2\right)^{\frac{3}{2}-2}\left(x\right)^{\frac{5}{2}}\left(y\right)^{-2+\frac{1}{8}}$$

   $$= \frac{\left(x\right)^{\frac{5}{2}}}{\left(2\right)^{\frac{1}{2}}\left(y\right)^{\frac{15}{8}}}$$

2. Answer:

   $$\left(5\sqrt{x^5}\right)\left(3\sqrt[3]{x}\right) = \left(5x^{\frac{5}{2}}\right)\left(3x^{\frac{1}{3}}\right)$$

   $$= 15x^{\frac{5}{2}+\frac{1}{3}}$$

   $$= 15x^{\frac{15+2}{6}}$$

   $$= 15x^{\frac{17}{6}}$$

   $$= 15x^{2\frac{5}{6}}$$

   $$= 15x^2\sqrt[6]{x^5}$$

3. Answer:

   $$\left(\frac{p^5q^{-10}}{p^5q^{-4}}\right)^{\frac{1}{2}}\left(\frac{p^{\frac{1}{4}}q^{-\frac{1}{2}}}{p^{-\frac{1}{2}}q^{-\frac{1}{2}}}\right)^{\frac{1}{2}} = \left(\frac{p^{5-5}q^{-10-(-4)}}{1}\right)^{\frac{1}{2}}\left(\frac{p^{\frac{1}{4}-(-\frac{1}{2})}q^{-\frac{1}{2}-(-\frac{1}{2})}}{1}\right)^{\frac{1}{2}}$$

   $$= \left(p^0q^{-6}\right)^{\frac{1}{2}}\left(p^{\frac{3}{4}}q^0\right)^{\frac{1}{2}}$$

   $$= \left(p^0q^{-3}\right)\left(p^{\frac{3}{8}}\cdot 1\right)$$

   $$= \left(1 \cdot q^{-3}\right)\left(p^{\frac{3}{8}} \cdot 1\right)$$

   $$= \frac{p^{\frac{3}{8}}}{q^3}$$

   $$= \frac{\sqrt[8]{p^3}}{q^3}$$

Rationalization Exercises

1. Rationalize the following expression:

   $$\frac{2}{\sqrt{b^3}}$$

2. Rationalize the following expression:

   $$\frac{2}{\sqrt{3} + \sqrt{5}}$$

3. Rationalize the following expression:

   $$\frac{m}{\sqrt{m} + n}$$

Answer Key for Rationalization

1. Answer:

   $$\frac{2}{\sqrt{b^3}} = \frac{2}{\sqrt{b^3}} \times \frac{\sqrt{b}}{\sqrt{b}}$$

   $$= \frac{2\sqrt{b}}{\sqrt{b^4}}$$

   $$= \frac{2\sqrt{b}}{b^2}$$

2. Answer:

   $$\frac{2}{\sqrt{3} + \sqrt{5}} = \frac{2}{\sqrt{3} + \sqrt{5}} \times \frac{\sqrt{3} - \sqrt{5}}{\sqrt{3} - \sqrt{5}}$$

   $$= \frac{2(\sqrt{3} - \sqrt{5})}{(\sqrt{3})^2 - (\sqrt{5})^2}$$

   $$= \frac{2(\sqrt{3} - \sqrt{5})}{3 - 5}$$

   $$= \frac{2(\sqrt{3} - \sqrt{5})}{-2}$$

   $$= -(\sqrt{3} - \sqrt{5})$$

   $$= \sqrt{5} - \sqrt{3}$$

3. Answer:

   $$\frac{m}{\sqrt{m} + n} = \frac{m}{\sqrt{m} + n} \times \frac{\sqrt{m} - n}{\sqrt{m} - n}$$

   $$= \frac{m(\sqrt{m} - n)}{(\sqrt{m})^2 - n^2}$$

   $$= \frac{m(\sqrt{m} - n)}{m - n^2}$$

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