Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{\frac{\frac{1}{x}+\frac{1}{2x+1}}{4x}}{2x+1}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\frac{\frac{3x+1}{2x^2+x}}{4x}}{2x+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{3x+1}{8x^3+4x^2}}{2x+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{3x+1}{16x^4+16x^3+4x^2}\end{aligned} $$ | |
① | To add raitonal expressions, both fractions must have the same denominator. |
② | Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{3x+1}{2x^2+x} }{4x} & \xlongequal{\text{Step 1}} \frac{3x+1}{2x^2+x} \cdot \frac{\color{blue}{1}}{\color{blue}{4x}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 3x+1 \right) \cdot 1 }{ \left( 2x^2+x \right) \cdot 4x } \xlongequal{\text{Step 3}} \frac{ 3x+1 }{ 8x^3+4x^2 } \end{aligned} $$ |
③ | Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{3x+1}{8x^3+4x^2} }{2x+1} & \xlongequal{\text{Step 1}} \frac{3x+1}{8x^3+4x^2} \cdot \frac{\color{blue}{1}}{\color{blue}{2x+1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 3x+1 \right) \cdot 1 }{ \left( 8x^3+4x^2 \right) \cdot \left( 2x+1 \right) } \xlongequal{\text{Step 3}} \frac{ 3x+1 }{ 16x^4+8x^3+8x^3+4x^2 } = \\[1ex] &= \frac{3x+1}{16x^4+16x^3+4x^2} \end{aligned} $$ |