Division of Radicals

\pi \cdot \pi

Division of Radicals

With division of radicals, we don't want radicals in the denominator.

For instance, in \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\), we don't want \(\sqrt[3]{2}\) in the denominator of the fraction.

We have various techniques to get rid of the radical in the denominator:

Rewrite the expression under a single radical and reduce the fraction.
Multiply the numerator and denominator by what is missing to make the radical a whole.
Rationalize the denominator by multiplying by the conjugate.

Division of Radicals

Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{54}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)

=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{9\sqrt[3]{2}}{4}\)

Division of Radicals

Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{54}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)

=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{9\sqrt[3]{2}}{4}\)

Division of Radicals

Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{54}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)

=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{9\sqrt[3]{2}}{4}\)

Division of Radicals

Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{54}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)

=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{9\sqrt[3]{2}}{4}\)

Division of Radicals

Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{54}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)

=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{9\sqrt[3]{2}}{4}\)

Division of Radicals

Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{54}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)

=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{9\sqrt[3]{2}}{4}\)

Division of Radicals

Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{54}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)

=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{9\sqrt[3]{2}}{4}\)

Division of Radicals

Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)

=\(\frac{15}{20} \cdot \sqrt[3]{54}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)

=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)

=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)

=\(\frac{9\sqrt[3]{2}}{4}\)

Division of Radicals

Ex 2: Simplify \(\frac{\sqrt{6}}{\sqrt{5}}\)

\(\frac{\sqrt{6}}{\sqrt{5}}\)

=\(\frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}\)

=\(\frac{\sqrt{6} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}\)

=\(\frac{\sqrt{6 \cdot 5}}{5}\)

=\(\frac{\sqrt{30}}{5}\)

Division of Radicals

Ex 3: Simplify \(\frac{\sqrt{6}}{\sqrt{5}}\)

\(\frac{\sqrt{6}}{\sqrt{5}}\)

=\(\frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}\)

=\(\frac{\sqrt{6} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}\)

=\(\frac{\sqrt{6 \cdot 5}}{5}\)

=\(\frac{\sqrt{30}}{5}\)

\pi \cdot \pi