Rationalize the Denominator with conjugates--Examples and interactive problems solved step by step

Rationalize the Denominator Worksheet

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

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 $$ 2 \red{-} \sqrt{5} $$ .

$$ \frac{3}{2 + \sqrt{5}} \red{ \frac{2 - \sqrt{5}} {2 - \sqrt{5}}} = \frac{3 (2 \red{-} \sqrt{5})} { (2\red{+} \sqrt{5})(2\red{-} \sqrt{5}) } \\ = \frac{ 6 -3\sqrt{5} } { 4 \red{+} 2\sqrt{5} \red{-} 2\sqrt{5} - 5 } = \frac{ 6 -3\sqrt{5} } { 4 \cancel { \red{+}2\sqrt{5} \red{-} 2\sqrt{5} } - 5 } \\ = \frac{ 6 -3\sqrt{5} } { 4 - 5 } = \frac{ 6 -3\sqrt{5} } { -1 } = \frac{ 6 -3\sqrt{5} } { -1 } $$

Step 3

Simplify.

$$ \frac{ 6 -3\sqrt{5} } { -1 } = 3\sqrt{5} - 6 $$

Problem 1

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 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

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.

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

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.