Section 10.3: Multiplying and Dividing Radical Expressions Worksheet 10.3
Fact of Life: Sometimes we need to multiply/divide radical expressions.
Recall:
$$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$$
$$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$
Example: Multiply and simplify the radical expression $\sqrt{13}(\sqrt{2}+\sqrt{7})$.
More Examples
$(9\sqrt{26 \eta})(2\sqrt{91 \eta})$
$6\sqrt{\tau}(8\sqrt{ \tau}-4)$
$8\sqrt[3]{3}(6\sqrt[3]{18}+7\sqrt[3]{45})$
$(\sqrt{2}+\sqrt{7})(\sqrt{2}-\sqrt{7})$ (hint: think FOIL)
$(\sqrt{3 \zeta} + \sqrt{11 \omega})^2$ (hint: think FOIL)
Rationalizing Denominators
Recall: Rationalizing denominators is simply rewriting a radical expression without
radicals in the denominator.
Example: $\frac{3}{\sqrt{2}}$
Example: $\frac{\sqrt{14}}{\sqrt{7}}$
Example: $\frac{15}{\sqrt{5 y}}$
Example: $\frac{\sqrt[3]{9}}{\sqrt[3]{16}}$
Rationalizing More Complicated Denominators
Fact of Life: sometimes denominators are a bit more complicated.
Example: Simplfy. $\frac{2}{5+\sqrt{7}}$
Conjugates: The conjugate of $5+\sqrt{7}$ is $5-\sqrt{7}.$
To find a conjugate, switch the middle sign.
Fact: to rationalize a denominator like the above, multiply top and bottom by
the conjugate of the denominator.
More Examples: Rationalize the denominators.
$\frac{3}{\sqrt{7}-\sqrt{2}}$
$\frac{\sqrt{p}}{\sqrt{\psi}-\sqrt{p}}$