Examples of rationalizing the denominator
Example 1: Conjugates
(more on rationalizing denominators with conjugates)Rationalize $$ \frac{3}{2 + \sqrt{5}} $$ Step 1
Multiply the numerator and denominator by the denominator's conjugate
$$ 2 \color{red}{-} \sqrt{5} $$Identify the conjugate of the denominator
$$
\frac{ 14}{ \color{red}{ 2 - \sqrt{7}} }
$$
Multiply the numerator and denominator by the conjugate
Identify the conjugate of the denominator
$$
\frac{ 9}{
\color{red}{ 3 + \sqrt{5} }
}
$$
Multiply the numerator and denominator by the conjugate
Identify the conjugate of the denominator
$$
\frac{ 3}{ \color{red}{\sqrt{7} +2 }}
$$
Identify the conjugate of the denominator
$$
\frac{ 5}{ \color{red}{ 3 - 2\sqrt{6}} }
$$
Multiply the numerator and denominator by the conjugate
Identify the conjugate of the denominator
$$
\frac{ 5}{ \color{red}{ 5 + 2\sqrt{7}} }
$$
Multiply the numerator and denominator by the conjugate
Challenge Problems
Identify the conjugate of the denominator
$$
\frac{ 7}{ \color{red}{\sqrt{2} + \sqrt{5}}}
$$
Multiply the numerator and denominator by the conjugate
Identify the conjugate of the denominator
$$
\frac{ \sqrt{7} -\sqrt{5} }{ \color{red}{ \sqrt{5} + \sqrt{7} }}
$$
Multiply the numerator and denominator by the conjugate
Rationalize Denominator Widget
Simply type into the app below and edit the expression. The Math Way app will solve it form there. You can visit this calculator on its own page here.
To read our review of the Math way--which is what fuels this page's calculator, please go here.