Question
Answer
$$\dfrac{30\sqrt{5}+70}{5+3\sqrt{5}}=3\sqrt{5}+5$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{30\sqrt{5}+70}{5+3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{30\sqrt{5}+70}{5+3\sqrt{5}}\frac{5-3\sqrt{5}}{5-3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{150\sqrt{5}-450+350-210\sqrt{5}}{25-15\sqrt{5}+15\sqrt{5}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-60\sqrt{5}-100}{-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-3\sqrt{5}-5}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{3\sqrt{5}+5}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}3\sqrt{5}+5\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 30 \sqrt{5} + 70\right) } \cdot \left( 5- 3 \sqrt{5}\right) = \color{blue}{ 30 \sqrt{5}} \cdot5+\color{blue}{ 30 \sqrt{5}} \cdot- 3 \sqrt{5}+\color{blue}{70} \cdot5+\color{blue}{70} \cdot- 3 \sqrt{5} = \\ = 150 \sqrt{5}-450 + 350- 210 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + 3 \sqrt{5}\right) } \cdot \left( 5- 3 \sqrt{5}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot5+\color{blue}{ 3 \sqrt{5}} \cdot- 3 \sqrt{5} = \\ = 25- 15 \sqrt{5} + 15 \sqrt{5}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 20. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
This page was created using
Rationalize Denominator Calculator