Question
Answer
$$\dfrac{12}{5\sqrt{3}-\sqrt{7}}=\dfrac{60\sqrt{3}+12\sqrt{7}}{68}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{12}{5\sqrt{3}-\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{12}{5\sqrt{3}-\sqrt{7}}\frac{5\sqrt{3}+\sqrt{7}}{5\sqrt{3}+\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{60\sqrt{3}+12\sqrt{7}}{75+5\sqrt{21}-5\sqrt{21}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{60\sqrt{3}+12\sqrt{7}}{68}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{3} + \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 12 } \cdot \left( 5 \sqrt{3} + \sqrt{7}\right) = \color{blue}{12} \cdot 5 \sqrt{3}+\color{blue}{12} \cdot \sqrt{7} = \\ = 60 \sqrt{3} + 12 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{3}- \sqrt{7}\right) } \cdot \left( 5 \sqrt{3} + \sqrt{7}\right) = \color{blue}{ 5 \sqrt{3}} \cdot 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot \sqrt{7}\color{blue}{- \sqrt{7}} \cdot 5 \sqrt{3}\color{blue}{- \sqrt{7}} \cdot \sqrt{7} = \\ = 75 + 5 \sqrt{21}- 5 \sqrt{21}-7 $$ |
| ③ | Simplify numerator and denominator |
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