Answer
$$\frac{7+5\sqrt{10}}{4-\sqrt{10}}=\frac{26+9\sqrt{10}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7+5\sqrt{10}}{4-\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7+5\sqrt{10}}{4-\sqrt{10}}\frac{4+\sqrt{10}}{4+\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{28+7\sqrt{10}+20\sqrt{10}+50}{16+4\sqrt{10}-4\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{78+27\sqrt{10}}{6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{26+9\sqrt{10}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7 + 5 \sqrt{10}\right) } \cdot \left( 4 + \sqrt{10}\right) = \color{blue}{7} \cdot4+\color{blue}{7} \cdot \sqrt{10}+\color{blue}{ 5 \sqrt{10}} \cdot4+\color{blue}{ 5 \sqrt{10}} \cdot \sqrt{10} = \\ = 28 + 7 \sqrt{10} + 20 \sqrt{10} + 50 $$ Simplify denominator. $$ \color{blue}{ \left( 4- \sqrt{10}\right) } \cdot \left( 4 + \sqrt{10}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{10}\color{blue}{- \sqrt{10}} \cdot4\color{blue}{- \sqrt{10}} \cdot \sqrt{10} = \\ = 16 + 4 \sqrt{10}- 4 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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