Answer
$$\frac{(4-\sqrt{5})\cdot(4+\sqrt{5})}{2\sqrt{11}}=\frac{\sqrt{11}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(4-\sqrt{5})\cdot(4+\sqrt{5})}{2\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{16+4\sqrt{5}-4\sqrt{5}-5}{2\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11}{2\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{11}{2\sqrt{11}}\frac{\sqrt{11}}{\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{11\sqrt{11}}{22} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{\sqrt{11}}{2}\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( 4- \sqrt{5}\right) } \cdot \left( 4 + \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot4\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 16 + 4 \sqrt{5}- 4 \sqrt{5}-5 $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{11}} $$. |
| ④ | Multiply in a numerator. $$ \color{blue}{ 11 } \cdot \sqrt{11} = 11 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ 2 \sqrt{11} } \cdot \sqrt{11} = 22 $$ |
| ⑤ | Divide both numerator and denominator by 11. |
This page was created using
Rationalize Denominator Calculator