RadF - Applying the Properties of Rational Exponents (Division) Lesson

Applying the Properties of Rational Exponents (Division)

Dividing Radicals

When multiplying radicals remember the following steps:

Divide the coefficients, and then use the Quotient Property of Radicals, which is $LaTeX: \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},\:b\ne0$ $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},\:b\ne0$
Simplify the resulting radical.

Rationalizing the Denominator

Often when performing operations on radical expressions, whether written in radical form or exponential form, the result will include a radical in the denominator. While the fraction may be in "simplest form," we typically do not write our final result with the denominator containing a radical. This is when we enlist a method called "rationalizing the denominator." This should be familiar to you as we have used this method before when we were dividing complex numbers and or using the quadratic formula to solve quadratic functions.

To " Rationalize the Denominator " containing a radical, remember the following points:

If the denominator contains a monomial, then multiply the numerator and denominator by the radical (or by a radical that would yield a result similar to $LaTeX: \frac{a}{\sqrt[m]{b^n}}\cdot\frac{\sqrt[m]{b^x}}{\frac{\sqrt[m]{b^x}}{ }}$ $\frac{a}{\sqrt[m]{b^n}}\cdot\frac{\sqrt[m]{b^x}}{\frac{\sqrt[m]{b^x}}{ }}$ , where n + x = m. This would yield a result of $LaTeX: a\cdot\frac{\left(\sqrt[m]{b^x}\right)}{\sqrt[m]{b^m}}=\frac{a\cdot\left(\sqrt[m]{b^x}\right)}{|b|}$ $a\cdot\frac{\left(\sqrt[m]{b^x}\right)}{\sqrt[m]{b^m}}=\frac{a\cdot\left(\sqrt[m]{b^x}\right)}{|b|}$ .
If the denominator contains a binomial, then multiply the numerator and denominator by the conjugate (the same expression but with the opposite sign between the two terms).

Examples: Write Radicals in Simplest Form

$LaTeX: \frac{\sqrt[5]{96}}{\sqrt[5]{3}} \\ \hspace{1cm}=\sqrt[5]{\frac{96}{5}}=\sqrt[5]{32}=2 \\ \frac{\sqrt[5]{7}}{\sqrt[5]{8}} \\ \hspace{1cm}=\frac{\sqrt[5]{7}}{\sqrt[5]{2^3}} \\ \hspace{1cm}=\frac{\sqrt[5]{7}}{\sqrt[5]{2^3}}\cdot\frac{\sqrt[5]{2^2}}{\sqrt[5]{2^2}} \\ \hspace{1cm}=\frac{\sqrt[5]{7\cdot2^2}}{\frac{\sqrt[5]{2^3\cdot2^2}}{ }}=\frac{\sqrt[5]{28}}{\sqrt[5]{32}} \\ \hspace{1cm}=\frac{\sqrt[5]{28}}{2}$ $\frac{\sqrt[5]{96}}{\sqrt[5]{3}} \\ \hspace{1cm}=\sqrt[5]{\frac{96}{5}}=\sqrt[5]{32}=2 \\ \frac{\sqrt[5]{7}}{\sqrt[5]{8}} \\ \hspace{1cm}=\frac{\sqrt[5]{7}}{\sqrt[5]{2^3}} \\ \hspace{1cm}=\frac{\sqrt[5]{7}}{\sqrt[5]{2^3}}\cdot\frac{\sqrt[5]{2^2}}{\sqrt[5]{2^2}} \\ \hspace{1cm}=\frac{\sqrt[5]{7\cdot2^2}}{\frac{\sqrt[5]{2^3\cdot2^2}}{ }}=\frac{\sqrt[5]{28}}{\sqrt[5]{32}} \\ \hspace{1cm}=\frac{\sqrt[5]{28}}{2}$

$LaTeX: \frac{3-\sqrt[4]{5k^2}}{\sqrt[4]{3k^3}} \\ \hspace{1cm}=\frac{\left(3-\sqrt[4]{5k^2}\right)}{\sqrt[4]{3k^3}}\cdot\frac{\sqrt[4]{3^3\cdot k^1}}{\sqrt[4]{3^3\cdot k^1}} \\ \hspace{1cm}=\frac{3\cdot\sqrt[4]{3^3\cdot k^1}-\sqrt[4]{5k^2}\cdot\sqrt[4]{3^3\cdot k^1}}{\sqrt[4]{3^4\cdot k^4}} \\ \hspace{1cm}=\frac{\sqrt[4]{27\cdot k}-\sqrt[4]{5\cdot3^3\cdot k^2\cdot k^1}}{\sqrt[4]{3^4\cdot k^4}} \\ \hspace{1cm}=\frac{3\cdot\sqrt[4]{27\cdot k}-\sqrt[4]{135k^3}}{3k} \\ \hspace{1cm}=\frac{3\sqrt[4]{27\cdot k}-\sqrt[4]{135k^3}}{3k}$ $\frac{3-\sqrt[4]{5k^2}}{\sqrt[4]{3k^3}} \\ \hspace{1cm}=\frac{\left(3-\sqrt[4]{5k^2}\right)}{\sqrt[4]{3k^3}}\cdot\frac{\sqrt[4]{3^3\cdot k^1}}{\sqrt[4]{3^3\cdot k^1}} \\ \hspace{1cm}=\frac{3\cdot\sqrt[4]{3^3\cdot k^1}-\sqrt[4]{5k^2}\cdot\sqrt[4]{3^3\cdot k^1}}{\sqrt[4]{3^4\cdot k^4}} \\ \hspace{1cm}=\frac{\sqrt[4]{27\cdot k}-\sqrt[4]{5\cdot3^3\cdot k^2\cdot k^1}}{\sqrt[4]{3^4\cdot k^4}} \\ \hspace{1cm}=\frac{3\cdot\sqrt[4]{27\cdot k}-\sqrt[4]{135k^3}}{3k} \\ \hspace{1cm}=\frac{3\sqrt[4]{27\cdot k}-\sqrt[4]{135k^3}}{3k}$

$LaTeX: \frac{8-\sqrt[]{6}}{3-4\sqrt[]{6}} \\ \hspace{1cm}=\frac{\left(8-\sqrt[]{6}\right)}{\left(3-4\sqrt[]{6}\right)}\cdot\frac{\left(3+4\sqrt[]{6}\right)}{\left(3+4\sqrt[]{6}\right)} \\ \hspace{1cm}=\frac{24+32\sqrt[]{6}-3\sqrt[]{6}-4\sqrt[]{6^2}}{9+12\sqrt[]{6}-12\sqrt[]{6}-16\sqrt[]{6^2}} \\ \hspace{1cm}=\frac{24+\left(32-3\right)\sqrt[]{6}-4\cdot6}{9+0-16\cdot6}=\frac{24-24+\left(32-3\right)\sqrt[]{6}}{9-96} \\ \hspace{1cm}=-\frac{29\sqrt[]{6}}{87}=-\frac{29}{87}\cdot\sqrt[]{6}=-\frac{1}{3}\cdot\sqrt[]{6}=\frac{\sqrt[]{6}}{3}$ $\frac{8-\sqrt[]{6}}{3-4\sqrt[]{6}} \\ \hspace{1cm}=\frac{\left(8-\sqrt[]{6}\right)}{\left(3-4\sqrt[]{6}\right)}\cdot\frac{\left(3+4\sqrt[]{6}\right)}{\left(3+4\sqrt[]{6}\right)} \\ \hspace{1cm}=\frac{24+32\sqrt[]{6}-3\sqrt[]{6}-4\sqrt[]{6^2}}{9+12\sqrt[]{6}-12\sqrt[]{6}-16\sqrt[]{6^2}} \\ \hspace{1cm}=\frac{24+\left(32-3\right)\sqrt[]{6}-4\cdot6}{9+0-16\cdot6}=\frac{24-24+\left(32-3\right)\sqrt[]{6}}{9-96} \\ \hspace{1cm}=-\frac{29\sqrt[]{6}}{87}=-\frac{29}{87}\cdot\sqrt[]{6}=-\frac{1}{3}\cdot\sqrt[]{6}=\frac{\sqrt[]{6}}{3}$

$LaTeX: \frac{\left(3-\sqrt[]{7}\right)}{\left(2-2\sqrt[]{7}\right)} \\ \hspace{1cm}=\frac{\left(3-\sqrt[]{7}\right)}{\left(2-2\sqrt[]{7}\right)}\cdot\frac{\left(2+2\sqrt[]{7}\right)}{\left(2+2\sqrt[]{7}\right)} \\ \hspace{1cm}=\frac{6+6\sqrt[]{7}-2\sqrt[]{7}-2\sqrt[]{7^2}}{4+4\sqrt[]{6}-4\sqrt[]{6}-4\sqrt[]{7^2}} \\ \hspace{1cm}=\frac{6+\left(6-2\right)\sqrt[]{7}-2\cdot7}{4+0-4\cdot7}=\frac{6-14+\left(4\right)\sqrt[]{6}}{4-28} \\ \hspace{1cm}=\frac{-8}{-24}+\frac{4\sqrt[]{7}}{-24}=\frac{1}{3}-\frac{\sqrt[]{7}}{6}=\frac{1}{3}-\frac{\sqrt[]{7}}{6}$ $\frac{\left(3-\sqrt[]{7}\right)}{\left(2-2\sqrt[]{7}\right)} \\ \hspace{1cm}=\frac{\left(3-\sqrt[]{7}\right)}{\left(2-2\sqrt[]{7}\right)}\cdot\frac{\left(2+2\sqrt[]{7}\right)}{\left(2+2\sqrt[]{7}\right)} \\ \hspace{1cm}=\frac{6+6\sqrt[]{7}-2\sqrt[]{7}-2\sqrt[]{7^2}}{4+4\sqrt[]{6}-4\sqrt[]{6}-4\sqrt[]{7^2}} \\ \hspace{1cm}=\frac{6+\left(6-2\right)\sqrt[]{7}-2\cdot7}{4+0-4\cdot7}=\frac{6-14+\left(4\right)\sqrt[]{6}}{4-28} \\ \hspace{1cm}=\frac{-8}{-24}+\frac{4\sqrt[]{7}}{-24}=\frac{1}{3}-\frac{\sqrt[]{7}}{6}=\frac{1}{3}-\frac{\sqrt[]{7}}{6}$

Dividing Radical Expressions Practice

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