Rationalize the Denominator with Conjugates

Examples, formula and the Steps!

Rationalize the Denominator Worksheet
(25 question worksheet with answer key)
Radicals Home

$$ \sqrt{45 } = \text{?} $$

Rationalize denominators

Examples of rationalizing the denominator

Example 1: Conjugates
(more on rationalizing denominators with conjugates)

Rationalize $$ \frac{3}{2 + \sqrt{5}} $$ Step 1

Simplify the square root in the denominator (if possible)
$$ \sqrt{5} $$ 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
$$ \frac{ 6 -3\sqrt{5} } { 9 } = \frac{ \color{red}{3}(2 -1\sqrt{5}) } { \color{red}{3}(3) } \\ \frac{ \color{red}{\cancel{3}}(2 -1\sqrt{5} ) } { \color{red}{\cancel{3} } (3 ) } = \frac{ (2 - \sqrt{5} )} { 3 } $$
Problem 1

Rationalize $$ \frac{ 14}{ 2 - \sqrt{7} } $$

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
Simplify
$$ \frac{ 14}{ 2 - \sqrt{7} } \cdot \frac{2 \color{red}{+} \sqrt{7} }{2 \color{red}{ + } \sqrt{7} } \\ = \frac{ 14(2 \color{red}{ + } \sqrt{7}) }{ (2 - \sqrt{7})(2 \color{red}{ + } \sqrt{7}) } \\ = \frac{ 28 + 14\sqrt{7} }{ 4 \color{red}{- 2\sqrt{7} + 2\sqrt{7}} -7 } \\ = \frac{ 28 + 14\sqrt{7} }{ -3 } $$
Problem 2

Rationalize $$ \frac{ 9}{ 3 + \sqrt{5} } $$

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
Simplify
$$ \frac{ 9}{ 3 + \sqrt{5} } \cdot \frac{ 3 \color{red}{ - } \sqrt{5} }{ 3 \color{red}{ - } \sqrt{5}} \\ \frac{ 9 (3 \color{red}{ - } \sqrt{5} ) }{ (3 + \sqrt{5})( 3 \color{red}{ - } \sqrt{5} )} \\ \frac{ 27 - 9\sqrt{5} } { 9 \color{red}{+3 \sqrt{5} -3 \sqrt{5} } - 5 } \\ \\ \frac{ 27 - 9\sqrt{5} } { 4 } $$
Problem 3

Rationalize $$ \frac{ 3}{ \sqrt{7} +2 } $$

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
Simplify
$$ \frac{ 3}{ \sqrt{7} +2 } \cdot \frac{ \sqrt{7} \color{red}{-} 2 }{ \sqrt{7} \color{red}{-} 2 } \\ \frac{ 3( \sqrt{7} \color{red}{-} 2) }{ (\sqrt{7} +2)(\sqrt{7} \color{red}{-} 2) } \\ \frac{ 3 \sqrt{7} \color{red}{-} 6 } {7 \color{red}{ +2\sqrt{7}-2\sqrt{7}} -4 } \\ \frac{ 3 \sqrt{7} \color{red}{-} 6 } {3} \\ \frac{ 3 ( \sqrt{7} \color{red}{-} 2) } {3(1)} \\ \frac{ \cancel{ 3} ( \sqrt{7} \color{red}{-} 2) } {\cancel{ 3}(1)} \\ \sqrt{7} \color{red}{-} 2 $$
Problem 4

Rationalize $$ \frac{ 5}{ 3 - 2\sqrt{6} } $$

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
Simplify
$$ \frac{ 5}{3 - 2\sqrt{6} } \cdot \frac{ 3 \color{red}{+} 2\sqrt{6} }{ 3 \color{red}{+} 2\sqrt{6} } \\ \frac{ 5( 3 \color{red}{+} 2\sqrt{6}) }{ (3 - 2\sqrt{6} ) ( 3 \color{red}{+} 2\sqrt{6})} \\ \frac { 15 + 10 \sqrt{6} }{ 9 \color{red}{ -6 \sqrt{6} +6\sqrt{6}} -24 } \\ \frac { 15 + 10 \sqrt{6} }{ -15 } \\ \frac { 15 + 10 \sqrt{6} }{ -15 } \\ \frac { 5(3 + 2 \sqrt{6}) }{ 5(-3) } \\ \frac { \cancel{5}(3 + 2 \sqrt{6}) }{ \cancel{5}(-3) } \\ \frac { 3 + 2 \sqrt{6} }{ -3 } $$
Problem 5

Rationalize $$ \frac{ 2\sqrt{3}}{ 5 + 2\sqrt{7} } $$

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
Simplify
$$ \frac{ 2\sqrt{3}}{ 5 + 2\sqrt{7} } \cdot \frac{5 \color{red}{-} 2\sqrt{ 7 } }{5 \color{red}{-} 2\sqrt{ 7 } } \\ \frac{ 2\sqrt{3}(5\color{red}{-} 2\sqrt{ 7 } ) }{ (5 + 2\sqrt{7 })( 5 \color{red}{-} 2\sqrt{ 7 } ) } \\ \frac{ 10\sqrt{3} - 4\sqrt{21} }{ 25 \color{red}{ +10\sqrt{ 7} -10 \sqrt{ 7} } + 28 } \\ \frac{ 10\sqrt{3} - 4\sqrt{21} }{ 53} $$

Challenge Problems

Problem 6

Rationalize $$ \frac{ 7}{ \sqrt{2} + \sqrt{5}} $$

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
Simplify
$$ \frac{ 7}{ \sqrt{2} + \sqrt{5}} \cdot \frac{ \sqrt{2} \color{red}{-} \sqrt{5} } { \sqrt{2} \color{red}{-} \sqrt{5}} \\ \frac{ 7( \sqrt{2} \color{red}{-} \sqrt{5} )} { ( \sqrt{2} + \sqrt{5})( \sqrt{2} \color{red}{-} \sqrt{5})} \\ \frac{ 7 \sqrt{2} - 7\sqrt{5} } { \sqrt{2}\sqrt{2} \color{red}{ - \sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{5}} - \sqrt{5}\sqrt{5} } \\ \frac{ 7 \sqrt{2} - 7\sqrt{5} } { 2 \color{red}{ - \sqrt{10} + \sqrt{10}} - 5 } \\ \frac{ 7 \sqrt{2} - 7\sqrt{5} } { 2 - 5 } \\ \frac{ 7 \sqrt{2} - 7\sqrt{5} } { -3} $$
Problem 7

Rationalize $$ \frac{ \sqrt{7} -\sqrt{5} }{ \sqrt{5} + \sqrt{7}} $$

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
Simplify
$$ \frac{ \sqrt{7} -\sqrt{5} }{ \sqrt{5} + \sqrt{7} } \cdot \frac{ \sqrt{5} \color{red}{-} \sqrt{7}} { \sqrt{5} \color{red}{-} \sqrt{7} } \\ \frac{ (\sqrt{7} -\sqrt{5})( \sqrt{5} \color{red}{-} \sqrt{7})} { ( \sqrt{5} + \sqrt{7})( \sqrt{5} \color{red}{-} \sqrt{7} )} \\ \frac{ \sqrt{7} \sqrt{5} -\sqrt{7}\sqrt{7} -\sqrt{5}\sqrt{5} + \sqrt{5}\sqrt{7} } { \sqrt{5} \sqrt{5} \color{red}{ -\sqrt{5}\sqrt{7}+\sqrt{5}\sqrt{7} }-\sqrt{7}\sqrt{7}} \\ \frac{ \sqrt{35} -7 -5 + \sqrt{35} } { 5 -7} \\ \frac{ -12 + 2 \sqrt{35} } { -2} \\ \frac{-2( 6 -1 \sqrt{35}) } { -2 (1)} \\ \frac{\cancel{-2}( 6 -1 \sqrt{35}) } { \cancel{-2 }(1)} \\ 6 - \sqrt{35} $$

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