Answer
$$\frac{21}{5\sqrt{10}+8\sqrt{5}}=\frac{-105\sqrt{10}+168\sqrt{5}}{70}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{21}{5\sqrt{10}+8\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{21}{5\sqrt{10}+8\sqrt{5}}\frac{5\sqrt{10}-8\sqrt{5}}{5\sqrt{10}-8\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{105\sqrt{10}-168\sqrt{5}}{250-200\sqrt{2}+200\sqrt{2}-320} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{105\sqrt{10}-168\sqrt{5}}{-70} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-105\sqrt{10}+168\sqrt{5}}{70}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{10}- 8 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 21 } \cdot \left( 5 \sqrt{10}- 8 \sqrt{5}\right) = \color{blue}{21} \cdot 5 \sqrt{10}+\color{blue}{21} \cdot- 8 \sqrt{5} = \\ = 105 \sqrt{10}- 168 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{10} + 8 \sqrt{5}\right) } \cdot \left( 5 \sqrt{10}- 8 \sqrt{5}\right) = \color{blue}{ 5 \sqrt{10}} \cdot 5 \sqrt{10}+\color{blue}{ 5 \sqrt{10}} \cdot- 8 \sqrt{5}+\color{blue}{ 8 \sqrt{5}} \cdot 5 \sqrt{10}+\color{blue}{ 8 \sqrt{5}} \cdot- 8 \sqrt{5} = \\ = 250- 200 \sqrt{2} + 200 \sqrt{2}-320 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
This page was created using
Rationalize Denominator Calculator