Question
Answer
$$\dfrac{32\sqrt{17}+51}{68+24\sqrt{17}}=\dfrac{-14\sqrt{17}+141}{76}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{32\sqrt{17}+51}{68+24\sqrt{17}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{32\sqrt{17}+51}{68+24\sqrt{17}}\frac{68-24\sqrt{17}}{68-24\sqrt{17}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2176\sqrt{17}-13056+3468-1224\sqrt{17}}{4624-1632\sqrt{17}+1632\sqrt{17}-9792} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{952\sqrt{17}-9588}{-5168} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{14\sqrt{17}-141}{-76} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-14\sqrt{17}+141}{76}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 68- 24 \sqrt{17}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 32 \sqrt{17} + 51\right) } \cdot \left( 68- 24 \sqrt{17}\right) = \color{blue}{ 32 \sqrt{17}} \cdot68+\color{blue}{ 32 \sqrt{17}} \cdot- 24 \sqrt{17}+\color{blue}{51} \cdot68+\color{blue}{51} \cdot- 24 \sqrt{17} = \\ = 2176 \sqrt{17}-13056 + 3468- 1224 \sqrt{17} $$ Simplify denominator. $$ \color{blue}{ \left( 68 + 24 \sqrt{17}\right) } \cdot \left( 68- 24 \sqrt{17}\right) = \color{blue}{68} \cdot68+\color{blue}{68} \cdot- 24 \sqrt{17}+\color{blue}{ 24 \sqrt{17}} \cdot68+\color{blue}{ 24 \sqrt{17}} \cdot- 24 \sqrt{17} = \\ = 4624- 1632 \sqrt{17} + 1632 \sqrt{17}-9792 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 68. |
| ⑤ | Multiply both numerator and denominator by -1. |
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