In this topic, multiplication will primarily involve binomials with radical terms. For example, something like: $$\left(\sqrt{5}-3\right)\left(\sqrt{6}+1\right)$$ Binomial means two terms, and radicals could be involved as terms. To handle these multiplication problems, think about the Distributive Property, or the FOIL method. I prefer the Distributive Property, so all the examples will follow that model.
| Example Multiply \(\left(7-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{2}\right)\) | Example Multiply \(\left(\sqrt[3]{7}+4\right)\left(\sqrt[3]{7}-4\right)\) |
The process of dividing by radicals is called rationalizing the denominator. Just like all division, the problem becomes multiplication by the correct radical expression. Depending on how complicated the denominator is, this correct radical expression needs to contain certain radicals for cancelling to occur. Each example that follows shows a different type of denominator.
Example (A single radical in the denominator)
Rationalize the denominator: \(\Large{\frac{8}{\sqrt{5}}}\)
Multiply by \(\Large{\frac{\sqrt{5}}{\sqrt{5}}\Rightarrow\frac{8}{\sqrt{5}}\cdot\color{red}{\frac{\sqrt{5}}{\sqrt{5}}}\Rightarrow \frac{8\sqrt{5}}{\sqrt{25}}\Rightarrow\frac{8\sqrt{5}}{5}}\). DO NOT CANCEL the 5s! One is inside the radical and one is outside.
The previous example was straight forward since is was square roots. The next one will have an index of 3 and show you how to handle an index greater than 2.
Example (A single radical of index greater than 2)
Rationalize the denominator: \(\Large{\sqrt[3]{\frac{15}{32}}}\).
| Apply Quotient Rule & Simplify: | \(\Large{\frac{\sqrt[3]{15}}{\sqrt[3]{32}}\Rightarrow\frac{\sqrt[3]{15}}{\sqrt[3]{8\cdot4}}\Rightarrow\frac{\sqrt[3]{15}}{\sqrt[3]{8}\cdot\sqrt[3]{4}}\Rightarrow\frac{\sqrt[3]{15}}{2\sqrt[3]{4}}}\) |
| Rationalize by \(\Large\frac{\sqrt[3]{2}}{\sqrt[3]{2}}\): | \(\Large{\frac{\sqrt[3]{15}}{2\sqrt[3]{4}}\cdot\color{red}{\frac{\sqrt[3]{2}}{\sqrt[3]{2}}}\Rightarrow\frac{\sqrt[3]{15\cdot2}}{\sqrt[3]{8}}\Rightarrow\frac{\sqrt[3]{30}}{2}}\) |
Further simplification of the numerator may be need, in this case \(\sqrt[3]{30}\) is completely simplified.
Note: the rationalization multiplier involved, \(\sqrt[3]{2}\), is used since a perfect cube will be the result after multiplying in the denominator. In this case, \(\sqrt[3]{8}\) is the result after multiplying and equals \(2\).
In this final example, the denominator is two terms instead of just one. Again, terms could be any radical, but for simplicity, this example shows square roots only.
Example (Two terms in the denominator)
Divide and simplify: \(\Large{\frac{-4}{\sqrt{5}+2}}\)