Answer
$$\frac{7\sqrt{5}-3\sqrt{2}}{7\sqrt{5}+3\sqrt{2}}=\frac{263-42\sqrt{10}}{227}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7\sqrt{5}-3\sqrt{2}}{7\sqrt{5}+3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7\sqrt{5}-3\sqrt{2}}{7\sqrt{5}+3\sqrt{2}}\frac{7\sqrt{5}-3\sqrt{2}}{7\sqrt{5}-3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{245-21\sqrt{10}-21\sqrt{10}+18}{245-21\sqrt{10}+21\sqrt{10}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{263-42\sqrt{10}}{227}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 \sqrt{5}- 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7 \sqrt{5}- 3 \sqrt{2}\right) } \cdot \left( 7 \sqrt{5}- 3 \sqrt{2}\right) = \color{blue}{ 7 \sqrt{5}} \cdot 7 \sqrt{5}+\color{blue}{ 7 \sqrt{5}} \cdot- 3 \sqrt{2}\color{blue}{- 3 \sqrt{2}} \cdot 7 \sqrt{5}\color{blue}{- 3 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 245- 21 \sqrt{10}- 21 \sqrt{10} + 18 $$ Simplify denominator. $$ \color{blue}{ \left( 7 \sqrt{5} + 3 \sqrt{2}\right) } \cdot \left( 7 \sqrt{5}- 3 \sqrt{2}\right) = \color{blue}{ 7 \sqrt{5}} \cdot 7 \sqrt{5}+\color{blue}{ 7 \sqrt{5}} \cdot- 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot 7 \sqrt{5}+\color{blue}{ 3 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 245- 21 \sqrt{10} + 21 \sqrt{10}-18 $$ |
| ③ | Simplify numerator and denominator |
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