Answer
$$\frac{(4-\sqrt{3})\cdot(4+\sqrt{3})}{\sqrt{13}}=\sqrt{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(4-\sqrt{3})\cdot(4+\sqrt{3})}{\sqrt{13}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{16+4\sqrt{3}-4\sqrt{3}-3}{\sqrt{13}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{13}{\sqrt{13}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}} \frac{ 13 }{\sqrt{ 13 }} \times \frac{ \color{orangered}{\sqrt{ 13 }} }{ \color{orangered}{\sqrt{ 13 }}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{13\sqrt{13}}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}} \frac{ 13 \sqrt{ 13 } : \color{blue}{ 13 } }{ 13 : \color{blue}{ 13 } } \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{\sqrt{13}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }\sqrt{13}\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( 4- \sqrt{3}\right) } \cdot \left( 4 + \sqrt{3}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{3}\color{blue}{- \sqrt{3}} \cdot4\color{blue}{- \sqrt{3}} \cdot \sqrt{3} = \\ = 16 + 4 \sqrt{3}- 4 \sqrt{3}-3 $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply both top and bottom by $ \color{orangered}{ \sqrt{ 13 }}$. |
| ④ | In denominator we have $ \sqrt{ 13 } \cdot \sqrt{ 13 } = 13 $. |
| ⑤ | Divide both the top and bottom numbers by $ \color{blue}{ 13 }$. |
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