Answer
$$\frac{36\sqrt{31}+55}{132-15\sqrt{31}}=\frac{1859\sqrt{31}+8000}{3483}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{36\sqrt{31}+55}{132-15\sqrt{31}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{36\sqrt{31}+55}{132-15\sqrt{31}}\frac{132+15\sqrt{31}}{132+15\sqrt{31}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4752\sqrt{31}+16740+7260+825\sqrt{31}}{17424+1980\sqrt{31}-1980\sqrt{31}-6975} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5577\sqrt{31}+24000}{10449} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1859\sqrt{31}+8000}{3483}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 132 + 15 \sqrt{31}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 36 \sqrt{31} + 55\right) } \cdot \left( 132 + 15 \sqrt{31}\right) = \color{blue}{ 36 \sqrt{31}} \cdot132+\color{blue}{ 36 \sqrt{31}} \cdot 15 \sqrt{31}+\color{blue}{55} \cdot132+\color{blue}{55} \cdot 15 \sqrt{31} = \\ = 4752 \sqrt{31} + 16740 + 7260 + 825 \sqrt{31} $$ Simplify denominator. $$ \color{blue}{ \left( 132- 15 \sqrt{31}\right) } \cdot \left( 132 + 15 \sqrt{31}\right) = \color{blue}{132} \cdot132+\color{blue}{132} \cdot 15 \sqrt{31}\color{blue}{- 15 \sqrt{31}} \cdot132\color{blue}{- 15 \sqrt{31}} \cdot 15 \sqrt{31} = \\ = 17424 + 1980 \sqrt{31}- 1980 \sqrt{31}-6975 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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