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- Rationalize radical denominator
Rationalize denominator calculator
This calculator removes square roots from the denominator. The calculator rationalizes denominators containing one or two radical terms. For one radical term, the caluclator multiplies the numerator and denominator by the square root, and for two radical terms, it uses multiplication by the conjugate method. Additionally, the calculator shows all the steps and easy-to-understand explanations.
solution
$$\frac{1-\sqrt{2}}{1+\sqrt{2}}=-3+2\sqrt{2}$$explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1-\sqrt{2}}{1+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1-\sqrt{2}}{1+\sqrt{2}}\frac{1-\sqrt{2}}{1-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1-\sqrt{2}-\sqrt{2}+2}{1-\sqrt{2}+\sqrt{2}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{3-2\sqrt{2}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-3+2\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-3+2\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1- \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 1- \sqrt{2}\right) } \cdot \left( 1- \sqrt{2}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot- \sqrt{2}\color{blue}{- \sqrt{2}} \cdot1\color{blue}{- \sqrt{2}} \cdot- \sqrt{2} = \\ = 1- \sqrt{2}- \sqrt{2} + 2 $$ Simplify denominator. $$ \color{blue}{ \left( 1 + \sqrt{2}\right) } \cdot \left( 1- \sqrt{2}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot1+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 1- \sqrt{2} + \sqrt{2}-2 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
| ⑤ | Remove 1 from denominator. |
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examples
example 1:ex 1:
$$\frac{1}{\sqrt{8}}$$
example 2:ex 2:
$$\frac{\sqrt{2} + 1}{\sqrt{2} -1}$$
example 3:ex 3:
$$\frac{3\sqrt{2} - 2\sqrt{3}}{2\sqrt{3} + 3\sqrt{2}}$$
example 4:ex 4:
$$\frac{1}{\sqrt{21} + \sqrt{7} + 2\sqrt{3} + 2}$$
example 5:ex 5:
$$\frac{\sqrt{3} + \sqrt{2} + 1}{\sqrt{3} - \sqrt{2} + 1}$$
example 6:ex 6:
$$\frac{11 - \sqrt{6}}{\sqrt{6} - 6}$$
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