Answer
$$\frac{6\sqrt{3}-3\sqrt{5}\cdot5\sqrt{3}+3\sqrt{5}}{(5\sqrt{3})^2-(3\sqrt{5})^2}=\frac{2\sqrt{3}-5\sqrt{15}+\sqrt{5}}{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6\sqrt{3}-3\sqrt{5}\cdot5\sqrt{3}+3\sqrt{5}}{(5\sqrt{3})^2-(3\sqrt{5})^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{3}-3\sqrt{5}\cdot5\sqrt{3}+3\sqrt{5}}{75-45} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{6\sqrt{3}-3\sqrt{5}\cdot5\sqrt{3}+3\sqrt{5}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{3}-5\sqrt{15}+\sqrt{5}}{10}\end{aligned} $$ | |
| ① | $$ (5\sqrt{3})^2 =
5^{ 2 } \cdot \sqrt{3} ^ { 2 } =
5^{ 2 } \sqrt{3} ^2 =
5^{ 2 } \lvert 3 \rvert =
75 $$ |
| ② | $$ (3\sqrt{5})^2 =
3^{ 2 } \cdot \sqrt{5} ^ { 2 } =
3^{ 2 } \sqrt{5} ^2 =
3^{ 2 } \lvert 5 \rvert =
45 $$ |
| ③ | Divide both numerator and denominator by 3. |
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