Answer
$$\frac{\sqrt{10}}{3\sqrt{5}-4}=\frac{15\sqrt{2}+4\sqrt{10}}{29}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{10}}{3\sqrt{5}-4}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{10}}{3\sqrt{5}-4}\frac{3\sqrt{5}+4}{3\sqrt{5}+4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15\sqrt{2}+4\sqrt{10}}{45+12\sqrt{5}-12\sqrt{5}-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15\sqrt{2}+4\sqrt{10}}{29}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5} + 4} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{10} } \cdot \left( 3 \sqrt{5} + 4\right) = \color{blue}{ \sqrt{10}} \cdot 3 \sqrt{5}+\color{blue}{ \sqrt{10}} \cdot4 = \\ = 15 \sqrt{2} + 4 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5}-4\right) } \cdot \left( 3 \sqrt{5} + 4\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot4\color{blue}{-4} \cdot 3 \sqrt{5}\color{blue}{-4} \cdot4 = \\ = 45 + 12 \sqrt{5}- 12 \sqrt{5}-16 $$ |
| ③ | Simplify numerator and denominator |
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