Question
Answer
$$\dfrac{(6-\sqrt{5})\cdot(6+\sqrt{5})}{\sqrt{31}}=\sqrt{31}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(6-\sqrt{5})\cdot(6+\sqrt{5})}{\sqrt{31}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{36+6\sqrt{5}-6\sqrt{5}-5}{\sqrt{31}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{31}{\sqrt{31}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}} \frac{ 31 }{\sqrt{ 31 }} \times \frac{ \color{orangered}{\sqrt{ 31 }} }{ \color{orangered}{\sqrt{ 31 }}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{31\sqrt{31}}{31} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}} \frac{ 31 \sqrt{ 31 } : \color{blue}{ 31 } }{ 31 : \color{blue}{ 31 } } \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{\sqrt{31}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }\sqrt{31}\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( 6- \sqrt{5}\right) } \cdot \left( 6 + \sqrt{5}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot6\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 36 + 6 \sqrt{5}- 6 \sqrt{5}-5 $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply both top and bottom by $ \color{orangered}{ \sqrt{ 31 }}$. |
| ④ | In denominator we have $ \sqrt{ 31 } \cdot \sqrt{ 31 } = 31 $. |
| ⑤ | Divide both the top and bottom numbers by $ \color{blue}{ 31 }$. |
This page was created using
Rationalize Denominator Calculator