Question
Answer
$$\dfrac{8+\sqrt{5}}{3\sqrt{5}-2}=\dfrac{26\sqrt{5}+31}{41}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8+\sqrt{5}}{3\sqrt{5}-2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8+\sqrt{5}}{3\sqrt{5}-2}\frac{3\sqrt{5}+2}{3\sqrt{5}+2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{24\sqrt{5}+16+15+2\sqrt{5}}{45+6\sqrt{5}-6\sqrt{5}-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{26\sqrt{5}+31}{41}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5} + 2} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 8 + \sqrt{5}\right) } \cdot \left( 3 \sqrt{5} + 2\right) = \color{blue}{8} \cdot 3 \sqrt{5}+\color{blue}{8} \cdot2+\color{blue}{ \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot2 = \\ = 24 \sqrt{5} + 16 + 15 + 2 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5}-2\right) } \cdot \left( 3 \sqrt{5} + 2\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot2\color{blue}{-2} \cdot 3 \sqrt{5}\color{blue}{-2} \cdot2 = \\ = 45 + 6 \sqrt{5}- 6 \sqrt{5}-4 $$ |
| ③ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator