Answer
$$\frac{1}{4\sqrt{5}+4\sqrt{3}}=\frac{\sqrt{5}-\sqrt{3}}{8}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{4\sqrt{5}+4\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{4\sqrt{5}+4\sqrt{3}}\frac{4\sqrt{5}-4\sqrt{3}}{4\sqrt{5}-4\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{5}-4\sqrt{3}}{80-16\sqrt{15}+16\sqrt{15}-48} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{5}-4\sqrt{3}}{32} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{5}-\sqrt{3}}{8}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{5}- 4 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 4 \sqrt{5}- 4 \sqrt{3}\right) = \color{blue}{1} \cdot 4 \sqrt{5}+\color{blue}{1} \cdot- 4 \sqrt{3} = \\ = 4 \sqrt{5}- 4 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{5} + 4 \sqrt{3}\right) } \cdot \left( 4 \sqrt{5}- 4 \sqrt{3}\right) = \color{blue}{ 4 \sqrt{5}} \cdot 4 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot- 4 \sqrt{3}+\color{blue}{ 4 \sqrt{3}} \cdot 4 \sqrt{5}+\color{blue}{ 4 \sqrt{3}} \cdot- 4 \sqrt{3} = \\ = 80- 16 \sqrt{15} + 16 \sqrt{15}-48 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
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