Answer
$$\frac{(\sqrt{5}+\sqrt{3})\cdot(5-\sqrt{3})}{\sqrt{22}}=\frac{5\sqrt{110}-\sqrt{330}+5\sqrt{66}-3\sqrt{22}}{22}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(\sqrt{5}+\sqrt{3})\cdot(5-\sqrt{3})}{\sqrt{22}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{5}-\sqrt{15}+5\sqrt{3}-3}{\sqrt{22}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{5}-\sqrt{15}+5\sqrt{3}-3}{\sqrt{22}}\frac{\sqrt{22}}{\sqrt{22}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5\sqrt{110}-\sqrt{330}+5\sqrt{66}-3\sqrt{22}}{22}\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( \sqrt{5} + \sqrt{3}\right) } \cdot \left( 5- \sqrt{3}\right) = \color{blue}{ \sqrt{5}} \cdot5+\color{blue}{ \sqrt{5}} \cdot- \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot5+\color{blue}{ \sqrt{3}} \cdot- \sqrt{3} = \\ = 5 \sqrt{5}- \sqrt{15} + 5 \sqrt{3}-3 $$ |
| ② | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{22}} $$. |
| ③ | Multiply in a numerator. $$ \color{blue}{ \left( 5 \sqrt{5}- \sqrt{15} + 5 \sqrt{3}-3\right) } \cdot \sqrt{22} = \color{blue}{ 5 \sqrt{5}} \cdot \sqrt{22}\color{blue}{- \sqrt{15}} \cdot \sqrt{22}+\color{blue}{ 5 \sqrt{3}} \cdot \sqrt{22}\color{blue}{-3} \cdot \sqrt{22} = \\ = 5 \sqrt{110}- \sqrt{330} + 5 \sqrt{66}- 3 \sqrt{22} $$ Simplify denominator. $$ \color{blue}{ \sqrt{22} } \cdot \sqrt{22} = 22 $$ |
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