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Question

Answer

$$\dfrac{6+\dfrac{3}{4x}}{\dfrac{4}{8x}+\dfrac{1}{10x^2}}=\dfrac{120x^2+15x}{10x+2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{6+\frac{3}{4x}}{\frac{4}{8x}+\frac{1}{10x^2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{24x+3}{4x}}{\frac{40x^2+8x}{80x^3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1920x^4+240x^3}{160x^3+32x^2} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{120x^2+15x}{10x+2}\end{aligned} $$
①

Add $6$ and $ \dfrac{3}{4x} $ to get $ \dfrac{ \color{purple}{ 24x+3 } }{ 4x }$.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 4x }$.

$$ \begin{aligned} 6+ \frac{3}{4x} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} + \frac{3}{4x} = \frac{ 6 \cdot \color{blue}{ 4x }}{ 1 \cdot \color{blue}{ 4x }} + \frac{ 3 }{ 4x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 24x } }{ 4x } + \frac{ \color{purple}{ 3 } }{ 4x }=\frac{ \color{purple}{ 24x+3 } }{ 4x } \end{aligned} $$
②

Add $ \dfrac{4}{8x} $ and $ \dfrac{1}{10x^2} $ to get $ \dfrac{ \color{purple}{ 40x^2+8x } }{ 80x^3 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 10x^2 }$ and the second by $\color{blue}{ 8x }$.

$$ \begin{aligned} \frac{4}{8x} + \frac{1}{10x^2} & = \frac{ 4 \cdot \color{blue}{ 10x^2 }}{ 8x \cdot \color{blue}{ 10x^2 }} + \frac{ 1 \cdot \color{blue}{ 8x }}{ 10x^2 \cdot \color{blue}{ 8x }} = \\[1ex] &=\frac{ \color{purple}{ 40x^2 } }{ 80x^3 } + \frac{ \color{purple}{ 8x } }{ 80x^3 }=\frac{ \color{purple}{ 40x^2+8x } }{ 80x^3 } \end{aligned} $$
③

Divide $ \dfrac{24x+3}{4x} $ by $ \dfrac{40x^2+8x}{80x^3} $ to get $ \dfrac{ 1920x^4+240x^3 }{ 160x^3+32x^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.

$$ \begin{aligned} \frac{ \frac{24x+3}{4x} }{ \frac{\color{blue}{40x^2+8x}}{\color{blue}{80x^3}} } & \xlongequal{\text{Step 1}} \frac{24x+3}{4x} \cdot \frac{\color{blue}{80x^3}}{\color{blue}{40x^2+8x}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 24x+3 \right) \cdot 80x^3 }{ 4x \cdot \left( 40x^2+8x \right) } \xlongequal{\text{Step 3}} \frac{ 1920x^4+240x^3 }{ 160x^3+32x^2 } \end{aligned} $$
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