Question
Answer
$$\dfrac{4}{7\sqrt{5}-5\sqrt{7}}=\dfrac{28\sqrt{5}+20\sqrt{7}}{70}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{7\sqrt{5}-5\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{7\sqrt{5}-5\sqrt{7}}\frac{7\sqrt{5}+5\sqrt{7}}{7\sqrt{5}+5\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{28\sqrt{5}+20\sqrt{7}}{245+35\sqrt{35}-35\sqrt{35}-175} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{28\sqrt{5}+20\sqrt{7}}{70}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 \sqrt{5} + 5 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( 7 \sqrt{5} + 5 \sqrt{7}\right) = \color{blue}{4} \cdot 7 \sqrt{5}+\color{blue}{4} \cdot 5 \sqrt{7} = \\ = 28 \sqrt{5} + 20 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 7 \sqrt{5}- 5 \sqrt{7}\right) } \cdot \left( 7 \sqrt{5} + 5 \sqrt{7}\right) = \color{blue}{ 7 \sqrt{5}} \cdot 7 \sqrt{5}+\color{blue}{ 7 \sqrt{5}} \cdot 5 \sqrt{7}\color{blue}{- 5 \sqrt{7}} \cdot 7 \sqrt{5}\color{blue}{- 5 \sqrt{7}} \cdot 5 \sqrt{7} = \\ = 245 + 35 \sqrt{35}- 35 \sqrt{35}-175 $$ |
| ③ | Simplify numerator and denominator |
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