Question
Answer
$$\dfrac{\sqrt{7}+9}{3+2\sqrt{7}}=\dfrac{15\sqrt{7}-13}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{7}+9}{3+2\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{7}+9}{3+2\sqrt{7}}\frac{3-2\sqrt{7}}{3-2\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3\sqrt{7}-14+27-18\sqrt{7}}{9-6\sqrt{7}+6\sqrt{7}-28} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-15\sqrt{7}+13}{-19} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{15\sqrt{7}-13}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- 2 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{7} + 9\right) } \cdot \left( 3- 2 \sqrt{7}\right) = \color{blue}{ \sqrt{7}} \cdot3+\color{blue}{ \sqrt{7}} \cdot- 2 \sqrt{7}+\color{blue}{9} \cdot3+\color{blue}{9} \cdot- 2 \sqrt{7} = \\ = 3 \sqrt{7}-14 + 27- 18 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + 2 \sqrt{7}\right) } \cdot \left( 3- 2 \sqrt{7}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- 2 \sqrt{7}+\color{blue}{ 2 \sqrt{7}} \cdot3+\color{blue}{ 2 \sqrt{7}} \cdot- 2 \sqrt{7} = \\ = 9- 6 \sqrt{7} + 6 \sqrt{7}-28 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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