Answer
$$\frac{8}{3\sqrt{5}+5\sqrt{3}}=\frac{-24\sqrt{5}+40\sqrt{3}}{30}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8}{3\sqrt{5}+5\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8}{3\sqrt{5}+5\sqrt{3}}\frac{3\sqrt{5}-5\sqrt{3}}{3\sqrt{5}-5\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{24\sqrt{5}-40\sqrt{3}}{45-15\sqrt{15}+15\sqrt{15}-75} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{24\sqrt{5}-40\sqrt{3}}{-30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-24\sqrt{5}+40\sqrt{3}}{30}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5}- 5 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 8 } \cdot \left( 3 \sqrt{5}- 5 \sqrt{3}\right) = \color{blue}{8} \cdot 3 \sqrt{5}+\color{blue}{8} \cdot- 5 \sqrt{3} = \\ = 24 \sqrt{5}- 40 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5} + 5 \sqrt{3}\right) } \cdot \left( 3 \sqrt{5}- 5 \sqrt{3}\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot- 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot 3 \sqrt{5}+\color{blue}{ 5 \sqrt{3}} \cdot- 5 \sqrt{3} = \\ = 45- 15 \sqrt{15} + 15 \sqrt{15}-75 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
This page was created using
Rationalize Denominator Calculator