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Rationalizing Denominators Solutions

• Expressions containing a radical in the denominator are    not simplified   .
• Therefore, the denominator needs to be    rationalized   .
• Rationalization is the process of making an    irrational    denominator (terms with a radical)    rational   .
• The process requires    multiplying    the numerator and denominator by a factor of one, so that the radicand in the denominator becomes an exact root.
  • For square roots, multiply the numerator and denominator by the    radical   .
  • For radicals with an index other than two, multiply by the term $\underline{ \sqrt[d]{x^{d–n}} }$ so that an exact root is formed.

Example 1

Simplify. Rationalize the denominator.

$\frac{11}{2\sqrt{5}}$

Example 2

Simplify. Rationalize the denominator.

$\sqrt[5]{\frac{3q}{4r^3}}$

$\frac{\sqrt[5]{3q}}{\sqrt[5]{2^2{r}^3}}$

$$\begin{gathered} \frac{\sqrt[5]{3q}}{\sqrt[5]{2^2{r}^3}}\left(\frac{\sqrt[5]{2^3{r}^2}}{\sqrt[5]{2^3{r}^2}}\right) \\ \frac{\sqrt[5]{24q{r}^2}}{2r} \end{gathered}$$

Example 3

Simplify. Rationalize the denominator. Variables represent positive values. 

$\frac{12y}{\sqrt[3]{{\left(4x\right)}^2}}$

$$\begin{gathered} \frac{12y}{\sqrt[3]{{\left(4x\right)}^2}}\left(\frac{\sqrt[3]{4x}}{\sqrt[3]{4x}}\right) \\ \frac{{}^3\cancel{12}y \sqrt[3]{4x}}{\cancel{4}x} \\ \frac{3y \sqrt[3]{4x}}{x} \end{gathered}$$