Question
Answer
$$\dfrac{1}{\sqrt{11}+5\sqrt{5}}=\dfrac{-\sqrt{11}+5\sqrt{5}}{114}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{\sqrt{11}+5\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{\sqrt{11}+5\sqrt{5}}\frac{\sqrt{11}-5\sqrt{5}}{\sqrt{11}-5\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{11}-5\sqrt{5}}{11-5\sqrt{55}+5\sqrt{55}-125} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{11}-5\sqrt{5}}{-114} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{11}+5\sqrt{5}}{114}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{11}- 5 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( \sqrt{11}- 5 \sqrt{5}\right) = \color{blue}{1} \cdot \sqrt{11}+\color{blue}{1} \cdot- 5 \sqrt{5} = \\ = \sqrt{11}- 5 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{11} + 5 \sqrt{5}\right) } \cdot \left( \sqrt{11}- 5 \sqrt{5}\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot- 5 \sqrt{5}+\color{blue}{ 5 \sqrt{5}} \cdot \sqrt{11}+\color{blue}{ 5 \sqrt{5}} \cdot- 5 \sqrt{5} = \\ = 11- 5 \sqrt{55} + 5 \sqrt{55}-125 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
This page was created using
Rationalize Denominator Calculator