Answer
$$\frac{7\sqrt{3}+8\sqrt{2}}{7\sqrt{3}-8\sqrt{2}}=\frac{275+112\sqrt{6}}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7\sqrt{3}+8\sqrt{2}}{7\sqrt{3}-8\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7\sqrt{3}+8\sqrt{2}}{7\sqrt{3}-8\sqrt{2}}\frac{7\sqrt{3}+8\sqrt{2}}{7\sqrt{3}+8\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{147+56\sqrt{6}+56\sqrt{6}+128}{147+56\sqrt{6}-56\sqrt{6}-128} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{275+112\sqrt{6}}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 \sqrt{3} + 8 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7 \sqrt{3} + 8 \sqrt{2}\right) } \cdot \left( 7 \sqrt{3} + 8 \sqrt{2}\right) = \color{blue}{ 7 \sqrt{3}} \cdot 7 \sqrt{3}+\color{blue}{ 7 \sqrt{3}} \cdot 8 \sqrt{2}+\color{blue}{ 8 \sqrt{2}} \cdot 7 \sqrt{3}+\color{blue}{ 8 \sqrt{2}} \cdot 8 \sqrt{2} = \\ = 147 + 56 \sqrt{6} + 56 \sqrt{6} + 128 $$ Simplify denominator. $$ \color{blue}{ \left( 7 \sqrt{3}- 8 \sqrt{2}\right) } \cdot \left( 7 \sqrt{3} + 8 \sqrt{2}\right) = \color{blue}{ 7 \sqrt{3}} \cdot 7 \sqrt{3}+\color{blue}{ 7 \sqrt{3}} \cdot 8 \sqrt{2}\color{blue}{- 8 \sqrt{2}} \cdot 7 \sqrt{3}\color{blue}{- 8 \sqrt{2}} \cdot 8 \sqrt{2} = \\ = 147 + 56 \sqrt{6}- 56 \sqrt{6}-128 $$ |
| ③ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator