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
$$
2 \red{-} \sqrt{5}
$$
.
$$
\frac{3}{2 + \sqrt{5}} \red{ \frac{2 - \sqrt{5}} {2 - \sqrt{5}}}
= \frac{3 (2 \red{-} \sqrt{5})} { (2\red{+} \sqrt{5})(2\red{-} \sqrt{5}) }
\\
=
\frac{ 6 -3\sqrt{5} } {
4 \red{+} 2\sqrt{5} \red{-} 2\sqrt{5} - 5
}
=
\frac{ 6 -3\sqrt{5} } {
4 \cancel { \red{+}2\sqrt{5} \red{-} 2\sqrt{5} } - 5
}
\\ =
\frac{ 6 -3\sqrt{5} } {
4 - 5
}
=
\frac{ 6 -3\sqrt{5} } { -1 }
=
\frac{ 6 -3\sqrt{5} } { -1 }
$$
Step 3
Simplify.
$$
\frac{ 6 -3\sqrt{5} } { -1 }
= 3\sqrt{5} - 6
$$
Problem 1
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
$$
\frac{ 14}{ 2 - \sqrt{7} }
\cdot \frac{2 \color{red}{+} \sqrt{7} }{2 \color{red}{ + } \sqrt{7} }
\\
= \frac{ 14(2 \color{red}{ + } \sqrt{7}) }{ (2 - \sqrt{7})(2 \color{red}{ + } \sqrt{7}) }
\\
= \frac{ 28 + 14\sqrt{7} }{ 4 \color{red}{- 2\sqrt{7} + 2\sqrt{7}} -7 }
\\
= \frac{ 28 + 14\sqrt{7} }{ -3 }
$$
Problem 2
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
$$
\frac{ 9}{
3 + \sqrt{5} }
\cdot
\frac{ 3 \color{red}{ - } \sqrt{5} }{ 3 \color{red}{ - } \sqrt{5}}
\\
\frac{ 9 (3 \color{red}{ - } \sqrt{5}) }{ (3 + \sqrt{5})(3 \color{red}{ - } \sqrt{5})}
\\
\frac{ 27 - 9\sqrt{5} }
{ 9 \color{red}{+3 \sqrt{5} -3 \sqrt{5} } - 5 }
\\ \\
\frac{ 27 - 9\sqrt{5} }
{ 4 }
$$
Problem 3
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
$$
\frac{ 3}{ \sqrt{7} +2 }
\cdot
\frac{ \sqrt{7} \color{red}{-} 2 }{ \sqrt{7} \color{red}{-} 2 }
\\
\frac{ 3(\sqrt{7} \color{red}{-} 2) }{ (\sqrt{7} +2)(\sqrt{7} \color{red}{-} 2) }
\\ \frac{ 3 \sqrt{7} \color{red}{-} 6 }
{7 \color{red}{ +2\sqrt{7}-2\sqrt{7}} -4 }
\\ \frac{ 3 \sqrt{7} \color{red}{-} 6 }
{3}
\\ \frac{ 3 (\sqrt{7} \color{red}{-} 2) }
{3(1)}
\\ \frac{ \cancel{ 3} (\sqrt{7} \color{red}{-} 2) }
{\cancel{ 3}(1)}
\\
\sqrt{7} \color{red}{-} 2
$$
Problem 4
Step 1
Identify the conjugate of the denominator
$$
\frac{ 5}{ \color{red}{ 3 - 2\sqrt{6}} }
$$
.
$$ 3 \color{red}{+} 2\sqrt{6} $$
Step 2
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 3
$$
\frac{ 5}{3 - 2\sqrt{6} }
\cdot
\frac{ 3 \color{red}{+} 2\sqrt{6} }{ 3 \color{red}{+} 2\sqrt{6} }
\\
\frac{ 5(3 \color{red}{+} 2\sqrt{6}) }{ (3 - 2\sqrt{6}) (3 \color{red}{+} 2\sqrt{6})}
\\ \frac { 15 + 10 \sqrt{6} }{ 9 \color{red}{ -6 \sqrt{6} +6\sqrt{6}} -24 }
\\
\frac { 15 + 10 \sqrt{6} }{ -15 }
\\
\frac { 15 + 10 \sqrt{6} }{ -15 }
\\
\frac { 5(3 + 2 \sqrt{6}) }{ 5(-3) }
\\
\frac { \cancel{5}(3 + 2 \sqrt{6}) }{ \cancel{5}(-3) }
\\
\frac { 3 + 2 \sqrt{6} }{ -3 }
$$
Problem 5
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
$$
\frac{ 2\sqrt{3}}{ 5 + 2\sqrt{7} }
\cdot
\frac{5 \color{red}{-} 2\sqrt{ 7 } }{5 \color{red}{-} 2\sqrt{ 7 } }
\\
\frac{ 2\sqrt{3}(5\color{red}{-} 2\sqrt{ 7 }) }{ (5 + 2\sqrt{7 })(5 \color{red}{-} 2\sqrt{ 7 }) }
\\
\frac{
10\sqrt{3} - 4\sqrt{21}
}{ 25 \color{red}{ +10\sqrt{ 7} -10 \sqrt{ 7} } + 28 }
\\
\frac{
10\sqrt{3} - 4\sqrt{21}
}{ 53}
$$
Problem 6
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
$$
\frac{ 7}{ \sqrt{2} + \sqrt{5}}
\cdot
\frac{ \sqrt{2} \color{red}{-} \sqrt{5} }
{ \sqrt{2} \color{red}{-} \sqrt{5}}
\\
\frac{ 7(\sqrt{2} \color{red}{-} \sqrt{5})}
{ (\sqrt{2} + \sqrt{5})(\sqrt{2} \color{red}{-} \sqrt{5})}
\\
\frac{ 7 \sqrt{2} - 7\sqrt{5} }
{ \sqrt{2}\sqrt{2} \color{red}{ - \sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{5}} - \sqrt{5}\sqrt{5} }
\\
\frac{ 7 \sqrt{2} - 7\sqrt{5} }
{ 2 \color{red}{ - \sqrt{10} + \sqrt{10}} - 5 }
\\
\frac{ 7 \sqrt{2} - 7\sqrt{5} }
{ 2 - 5 }
\\
\frac{ 7 \sqrt{2} - 7\sqrt{5} }
{ -3}
$$
Problem 7
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
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