Answer
$$\frac{(3\sqrt{2}+7\sqrt{5})(2\sqrt{3}-6\sqrt{5})}{2\sqrt{3}\cdot6\sqrt{5}(2\sqrt{3}-6\sqrt{5})}=0$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(3\sqrt{2}+7\sqrt{5})(2\sqrt{3}-6\sqrt{5})}{2\sqrt{3}\cdot6\sqrt{5}(2\sqrt{3}-6\sqrt{5})}& \xlongequal{ }(6\sqrt{6}-18\sqrt{10}+14\sqrt{15}-210)\cdot\frac{1}{72\sqrt{5}-360\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6\sqrt{6}-18\sqrt{10}+14\sqrt{15}-210}{72\sqrt{5}-360\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{6}-18\sqrt{10}+14\sqrt{15}-210}{72\sqrt{5}-360\sqrt{3}}\frac{72\sqrt{5}+360\sqrt{3}}{72\sqrt{5}+360\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{432\sqrt{30}+6480\sqrt{2}-6480\sqrt{2}-6480\sqrt{30}+5040\sqrt{3}+15120\sqrt{5}-15120\sqrt{5}-75600\sqrt{3}}{25920+25920\sqrt{15}-25920\sqrt{15}-388800} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{0}{-362880} \xlongequal{ } \\[1 em] & \xlongequal{ }0\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( 6 \sqrt{6}- 18 \sqrt{10} + 14 \sqrt{15}-210\right) } \cdot 1 = \color{blue}{ 6 \sqrt{6}} \cdot1\color{blue}{- 18 \sqrt{10}} \cdot1+\color{blue}{ 14 \sqrt{15}} \cdot1\color{blue}{-210} \cdot1 = \\ = 6 \sqrt{6}- 18 \sqrt{10} + 14 \sqrt{15}-210 $$$$ \color{blue}{ 1 } \cdot \left( 72 \sqrt{5}- 360 \sqrt{3}\right) = \color{blue}{1} \cdot 72 \sqrt{5}+\color{blue}{1} \cdot- 360 \sqrt{3} = \\ = 72 \sqrt{5}- 360 \sqrt{3} $$ |
| ② | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 72 \sqrt{5} + 360 \sqrt{3}} $$. |
| ③ | Multiply in a numerator. $$ \color{blue}{ \left( 6 \sqrt{6}- 18 \sqrt{10} + 14 \sqrt{15}-210\right) } \cdot \left( 72 \sqrt{5} + 360 \sqrt{3}\right) = \color{blue}{ 6 \sqrt{6}} \cdot 72 \sqrt{5}+\color{blue}{ 6 \sqrt{6}} \cdot 360 \sqrt{3}\color{blue}{- 18 \sqrt{10}} \cdot 72 \sqrt{5}\color{blue}{- 18 \sqrt{10}} \cdot 360 \sqrt{3}+\color{blue}{ 14 \sqrt{15}} \cdot 72 \sqrt{5}+\color{blue}{ 14 \sqrt{15}} \cdot 360 \sqrt{3}\color{blue}{-210} \cdot 72 \sqrt{5}\color{blue}{-210} \cdot 360 \sqrt{3} = \\ = 432 \sqrt{30} + 6480 \sqrt{2}- 6480 \sqrt{2}- 6480 \sqrt{30} + 5040 \sqrt{3} + 15120 \sqrt{5}- 15120 \sqrt{5}- 75600 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 72 \sqrt{5}- 360 \sqrt{3}\right) } \cdot \left( 72 \sqrt{5} + 360 \sqrt{3}\right) = \color{blue}{ 72 \sqrt{5}} \cdot 72 \sqrt{5}+\color{blue}{ 72 \sqrt{5}} \cdot 360 \sqrt{3}\color{blue}{- 360 \sqrt{3}} \cdot 72 \sqrt{5}\color{blue}{- 360 \sqrt{3}} \cdot 360 \sqrt{3} = \\ = 25920 + 25920 \sqrt{15}- 25920 \sqrt{15}-388800 $$ |
| ④ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator