Examples of rationalizing the denominator
Example 1: Conjugates
(more on rationalizing denominators with conjugates)
Rationalize
$$
\frac{3}{2 + \sqrt{5}}
$$
Step 1
$$
\sqrt{5}
$$
is already simplified, so nothing to do here
Step 2
Multiply the numerator and denominator by the denominator's conjugate
$$
2 \color{red}{-} \sqrt{5}
$$
$$
\frac{3}{2 + \sqrt{5}} \color{red}{ \frac{2 - \sqrt{5}} {2 - \sqrt{5}}}
= \frac{3 (2 \color{red}{-} \sqrt{5} )} { (2\color{red}{+} \sqrt{5} )(2\color{red}{-} \sqrt{5} ) }
\\
=
\frac{ 6 -3\sqrt{5} } {
4 \color{red}{+} 2\sqrt{5} \color{red}{-} 2\sqrt{5}+ 5
}
=
\frac{ 6 -3\sqrt{5} } {
4 \cancel { \color{red}{+}2\sqrt{5} \color{red}{-} 2\sqrt{5} } + 5
}
\\ =
\frac{ 6 -3\sqrt{5} } {
4 + 5
}
=
\frac{ 6 -3\sqrt{5} } { 9 }
$$
Step 3
Simplify
$$
\frac{ 6 -3\sqrt{5} } { 9 }
=
\frac{ \color{red}{3}(2 -1\sqrt{5}) } { \color{red}{3}(3) }
\\
\frac{ \color{red}{\cancel{3}}(2 -1\sqrt{5} ) } { \color{red}{\cancel{3} } (3 ) }
= \frac{ (2 - \sqrt{5} )} { 3 }
$$
Step 1
Identify the conjugate of the denominator
$$
\frac{ 14}{ \color{red}{ 2 - \sqrt{7}} }
$$
$$ 2 \color{red}{+} \sqrt{7} $$
Step 2
Multiply the numerator and denominator by the conjugate
$$
\frac{ 14}{ \color{red}{2 - \sqrt{7}} }
\cdot \frac{2 \color{red}{+} \sqrt{7} }{2 \color{red}{ + } \sqrt{7} }
$$
Step 3
$$
\frac{ 14}{ 2 - \sqrt{7} }
\cdot \frac{2 \color{red}{+} \sqrt{7} }{2 \color{red}{ + } \sqrt{7} }
\\
= \frac{ 14(2 \color{red}{ + } \sqrt{7}) }{ (2 - \sqrt{7})(2 \color{red}{ + } \sqrt{7}) }
\\
= \frac{ 28 + 14\sqrt{7} }{ 4 \color{red}{- 2\sqrt{7} + 2\sqrt{7}} -7 }
\\
= \frac{ 28 + 14\sqrt{7} }{ -3 }
$$
Step 1
Identify the conjugate of the denominator
$$
\frac{ 9}{
\color{red}{ 3 + \sqrt{5} }
}
$$
$$
3 \color{red}{ -} \sqrt{5}
$$
Step 2
Multiply the numerator and denominator by the conjugate
$$
\frac{ 9}{
\color{red}{ 3 + \sqrt{5} }
}
\cdot
\frac{ 3 \color{red}{ - } \sqrt{5} }{ 3 \color{red}{ - } \sqrt{5}}
$$
Step 3
$$
\frac{ 9}{
3 + \sqrt{5} }
\cdot
\frac{ 3 \color{red}{ - } \sqrt{5} }{ 3 \color{red}{ - } \sqrt{5}}
\\
\frac{ 9 (3 \color{red}{ - } \sqrt{5} ) }{ (3 + \sqrt{5})( 3 \color{red}{ - } \sqrt{5} )}
\\
\frac{ 27 - 9\sqrt{5} }
{ 9 \color{red}{+3 \sqrt{5} -3 \sqrt{5} } - 5 }
\\
\\
\frac{ 27 - 9\sqrt{5} }
{ 4 }
$$
Step 1
Identify the conjugate of the denominator
$$
\frac{ 3}{ \color{red}{\sqrt{7} +2 }}
$$
$$
\sqrt{7} \color{red}{-} 2
$$
Step 2
Multiply the numerator and denominator by the conjugate
$$
\frac{ 3}{
\color{red}{\sqrt{7} +2 }}
\cdot
\frac{
\sqrt{7} \color{red}{-} 2 }{ \sqrt{7} \color{red}{-} 2 }
$$
Step 3
$$
\frac{ 3}{ \sqrt{7} +2 }
\cdot
\frac{ \sqrt{7} \color{red}{-} 2 }{ \sqrt{7} \color{red}{-} 2 }
\\
\frac{ 3( \sqrt{7} \color{red}{-} 2) }{ (\sqrt{7} +2)(\sqrt{7} \color{red}{-} 2) }
\\
\frac{ 3 \sqrt{7} \color{red}{-} 6 }
{7 \color{red}{ +2\sqrt{7}-2\sqrt{7}} -4 }
\\
\frac{ 3 \sqrt{7} \color{red}{-} 6 }
{3}
\\
\frac{ 3 ( \sqrt{7} \color{red}{-} 2) }
{3(1)}
\\
\frac{ \cancel{ 3} ( \sqrt{7} \color{red}{-} 2) }
{\cancel{ 3}(1)}
\\
\sqrt{7} \color{red}{-} 2
$$
Step 1
Identify the conjugate of the denominator
$$
\frac{ 5}{ \color{red}{ 3 - 2\sqrt{6}} }
$$
$$ 3 \color{red}{+} 2\sqrt{6} $$
Step 1
Multiply the numerator and denominator by the conjugate
$$
\frac{ 5}{ \color{red}{ 3 - 2\sqrt{6}} }
\cdot
\frac{ 3 \color{red}{+} 2\sqrt{6} }{ 3 \color{red}{+} 2\sqrt{6} }
$$
Step 1
$$
\frac{ 5}{3 - 2\sqrt{6} }
\cdot
\frac{ 3 \color{red}{+} 2\sqrt{6} }{ 3 \color{red}{+} 2\sqrt{6} }
\\
\frac{ 5( 3 \color{red}{+} 2\sqrt{6}) }{ (3 - 2\sqrt{6} ) ( 3 \color{red}{+} 2\sqrt{6})}
\\
\frac { 15 + 10 \sqrt{6} }{ 9 \color{red}{ -6 \sqrt{6} +6\sqrt{6}} -24 }
\\
\frac { 15 + 10 \sqrt{6} }{ -15 }
\\
\frac { 15 + 10 \sqrt{6} }{ -15 }
\\
\frac { 5(3 + 2 \sqrt{6}) }{ 5(-3) }
\\
\frac { \cancel{5}(3 + 2 \sqrt{6}) }{ \cancel{5}(-3) }
\\
\frac { 3 + 2 \sqrt{6} }{ -3 }
$$
Step 1
Identify the conjugate of the denominator
$$
\frac{ 5}{ \color{red}{ 5 + 2\sqrt{7}} }
$$
$$ 5 \color{red}{-} 2\sqrt{ 7 } $$
Step 2
Multiply the numerator and denominator by the conjugate
$$
\frac{ 2\sqrt{3}}{ \color{red} {5 + 2\sqrt{7}} }
\cdot
\frac{5 \color{red}{-} 2\sqrt{ 7 } }{5 \color{red}{-} 2\sqrt{ 7 } }
$$
Step 3
$$
\frac{ 2\sqrt{3}}{ 5 + 2\sqrt{7} }
\cdot
\frac{5 \color{red}{-} 2\sqrt{ 7 } }{5 \color{red}{-} 2\sqrt{ 7 } }
\\
\frac{ 2\sqrt{3}(5\color{red}{-} 2\sqrt{ 7 } ) }{ (5 + 2\sqrt{7 })( 5 \color{red}{-} 2\sqrt{ 7 } ) }
\\
\frac{
10\sqrt{3} - 4\sqrt{21}
}{ 25 \color{red}{ +10\sqrt{ 7} -10 \sqrt{ 7} } + 28 }
\\
\frac{
10\sqrt{3} - 4\sqrt{21}
}{ 53}
$$
Challenge Problems
Step 1
Identify the conjugate of the denominator
$$
\frac{ 7}{ \color{red}{\sqrt{2} + \sqrt{5}}}
$$
$$
\sqrt{2} \color{red}{-} \sqrt{5}
$$
Step 2
Multiply the numerator and denominator by the conjugate
$$
\frac{ 7}{ \sqrt{2} + \sqrt{5}}
\cdot
\frac{ \sqrt{2} \color{red}{-} \sqrt{5} }
{ \sqrt{2} \color{red}{-} \sqrt{5}}
$$
Step 3
$$
\frac{ 7}{ \sqrt{2} + \sqrt{5}}
\cdot
\frac{ \sqrt{2} \color{red}{-} \sqrt{5} }
{ \sqrt{2} \color{red}{-} \sqrt{5}}
\\
\frac{ 7( \sqrt{2} \color{red}{-} \sqrt{5} )}
{ ( \sqrt{2} + \sqrt{5})( \sqrt{2} \color{red}{-} \sqrt{5})}
\\
\frac{ 7 \sqrt{2} - 7\sqrt{5} }
{ \sqrt{2}\sqrt{2} \color{red}{ - \sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{5}} - \sqrt{5}\sqrt{5} }
\\
\frac{ 7 \sqrt{2} - 7\sqrt{5} }
{ 2 \color{red}{ - \sqrt{10} + \sqrt{10}} - 5 }
\\
\frac{ 7 \sqrt{2} - 7\sqrt{5} }
{ 2 - 5 }
\\
\frac{ 7 \sqrt{2} - 7\sqrt{5} }
{ -3}
$$
Step 1
Identify the conjugate of the denominator
$$
\frac{ \sqrt{7} -\sqrt{5} }{ \color{red}{ \sqrt{5} + \sqrt{7} }}
$$
$$
\sqrt{5} \color{red}{-} \sqrt{7}
$$
Step 2
Multiply the numerator and denominator by the conjugate
$$
\frac{ \sqrt{7} -\sqrt{5} }{ \color{red}{ \sqrt{5} + \sqrt{7} }}
\cdot
\frac{ \sqrt{5} \color{red}{-} \sqrt{7}}
{ \sqrt{5} \color{red}{-} \sqrt{7} }
$$
Step 3
$$
\frac{ \sqrt{7} -\sqrt{5} }{ \sqrt{5} + \sqrt{7} }
\cdot
\frac{ \sqrt{5} \color{red}{-} \sqrt{7}}
{ \sqrt{5} \color{red}{-} \sqrt{7} }
\\
\frac{ (\sqrt{7} -\sqrt{5})( \sqrt{5} \color{red}{-} \sqrt{7})}
{ ( \sqrt{5} + \sqrt{7})( \sqrt{5} \color{red}{-} \sqrt{7} )}
\\
\frac{ \sqrt{7} \sqrt{5} -\sqrt{7}\sqrt{7} -\sqrt{5}\sqrt{5} + \sqrt{5}\sqrt{7} }
{ \sqrt{5} \sqrt{5} \color{red}{ -\sqrt{5}\sqrt{7}+\sqrt{5}\sqrt{7} }-\sqrt{7}\sqrt{7}}
\\
\frac{ \sqrt{35} -7 -5 + \sqrt{35} }
{ 5 -7}
\\
\frac{ -12 + 2 \sqrt{35} }
{ -2}
\\
\frac{-2( 6 -1 \sqrt{35}) }
{ -2 (1)}
\\
\frac{\cancel{-2}( 6 -1 \sqrt{35}) }
{ \cancel{-2 }(1)}
\\
6 - \sqrt{35}
$$
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.
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