Question
Answer
$$\dfrac{3x^2(1+x^2+3y^4)-2x(x^3-y^2)}{(x^2+1+3y^4)^2}=\dfrac{9x^2y^4+x^4+2xy^2+3x^2}{9y^8+6x^2y^4+x^4+6y^4+2x^2+1}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3x^2(1+x^2+3y^4)-2x(x^3-y^2)}{(x^2+1+3y^4)^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3x^2+3x^4+9x^2y^4-(2x^4-2xy^2)}{(x^2+1+3y^4)^2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{3x^2+3x^4+9x^2y^4-2x^4+2xy^2}{9y^8+6x^2y^4+x^4+6y^4+2x^2+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{9x^2y^4+x^4+2xy^2+3x^2}{9y^8+6x^2y^4+x^4+6y^4+2x^2+1}\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{3x^2} $ by $ \left( 1+x^2+3y^4\right) $ $$ \color{blue}{3x^2} \cdot \left( 1+x^2+3y^4\right) = 3x^2+3x^4+9x^2y^4 $$ |
| ② | Multiply $ \color{blue}{2x} $ by $ \left( x^3-y^2\right) $ $$ \color{blue}{2x} \cdot \left( x^3-y^2\right) = 2x^4-2xy^2 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 2x^4-2xy^2 \right) = -2x^4+2xy^2 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^2+1+3y^4}\right) $ by each term in $ \left( x^2+1+3y^4\right) $. $$ \left( \color{blue}{x^2+1+3y^4}\right) \cdot \left( x^2+1+3y^4\right) = x^4+x^2+3x^2y^4+x^2+1+3y^4+3x^2y^4+3y^4+9y^8 $$ |
| ⑤ | Combine like terms: $$ x^4+ \color{blue}{x^2} + \color{red}{3x^2y^4} + \color{blue}{x^2} +1+ \color{green}{3y^4} + \color{red}{3x^2y^4} + \color{green}{3y^4} +9y^8 = \\ = 9y^8+ \color{red}{6x^2y^4} +x^4+ \color{green}{6y^4} + \color{blue}{2x^2} +1 $$ |
| ⑥ | Simplify numerator $$ 3x^2+ \color{blue}{3x^4} +9x^2y^4 \color{blue}{-2x^4} +2xy^2 = 9x^2y^4+ \color{blue}{x^4} +2xy^2+3x^2 $$ |
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