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Question

Answer

$$\dfrac{x}{1}+\dfrac{x^2}{3}+\dfrac{x^2}{5}+\dfrac{x^2}{7}+\dfrac{x^2}{9}+\dfrac{x^2}{11}+\dfrac{x^2}{13}=\dfrac{43024x^2+45045x}{45045}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{x}{1}+\frac{x^2}{3}+\frac{x^2}{5}+\frac{x^2}{7}+\frac{x^2}{9}+\frac{x^2}{11}+\frac{x^2}{13}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{x^2+3x}{3}+\frac{x^2}{5}+\frac{x^2}{7}+\frac{x^2}{9}+\frac{x^2}{11}+\frac{x^2}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8x^2+15x}{15}+\frac{x^2}{7}+\frac{x^2}{9}+\frac{x^2}{11}+\frac{x^2}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{71x^2+105x}{105}+\frac{x^2}{9}+\frac{x^2}{11}+\frac{x^2}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{248x^2+315x}{315}+\frac{x^2}{11}+\frac{x^2}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{3043x^2+3465x}{3465}+\frac{x^2}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{43024x^2+45045x}{45045}\end{aligned} $$
①

Add $ \dfrac{x}{1} $ and $ \dfrac{x^2}{3} $ to get $ \dfrac{ \color{purple}{ x^2+3x } }{ 3 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 3 }$.

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

Add $ \dfrac{x^2+3x}{3} $ and $ \dfrac{x^2}{5} $ to get $ \dfrac{ \color{purple}{ 8x^2+15x } }{ 15 }$.

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}{ 5 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{x^2+3x}{3} + \frac{x^2}{5} & = \frac{ \left( x^2+3x \right) \cdot \color{blue}{ 5 }}{ 3 \cdot \color{blue}{ 5 }} + \frac{ x^2 \cdot \color{blue}{ 3 }}{ 5 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 5x^2+15x } }{ 15 } + \frac{ \color{purple}{ 3x^2 } }{ 15 }=\frac{ \color{purple}{ 8x^2+15x } }{ 15 } \end{aligned} $$
③

Add $ \dfrac{8x^2+15x}{15} $ and $ \dfrac{x^2}{7} $ to get $ \dfrac{ \color{purple}{ 71x^2+105x } }{ 105 }$.

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}{ 7 }$ and the second by $\color{blue}{ 15 }$.

$$ \begin{aligned} \frac{8x^2+15x}{15} + \frac{x^2}{7} & = \frac{ \left( 8x^2+15x \right) \cdot \color{blue}{ 7 }}{ 15 \cdot \color{blue}{ 7 }} + \frac{ x^2 \cdot \color{blue}{ 15 }}{ 7 \cdot \color{blue}{ 15 }} = \\[1ex] &=\frac{ \color{purple}{ 56x^2+105x } }{ 105 } + \frac{ \color{purple}{ 15x^2 } }{ 105 }=\frac{ \color{purple}{ 71x^2+105x } }{ 105 } \end{aligned} $$
④

Add $ \dfrac{71x^2+105x}{105} $ and $ \dfrac{x^2}{9} $ to get $ \dfrac{ \color{purple}{ 248x^2+315x } }{ 315 }$.

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}{ 3 }$ and the second by $\color{blue}{ 35 }$.

$$ \begin{aligned} \frac{71x^2+105x}{105} + \frac{x^2}{9} & = \frac{ \left( 71x^2+105x \right) \cdot \color{blue}{ 3 }}{ 105 \cdot \color{blue}{ 3 }} + \frac{ x^2 \cdot \color{blue}{ 35 }}{ 9 \cdot \color{blue}{ 35 }} = \\[1ex] &=\frac{ \color{purple}{ 213x^2+315x } }{ 315 } + \frac{ \color{purple}{ 35x^2 } }{ 315 }=\frac{ \color{purple}{ 248x^2+315x } }{ 315 } \end{aligned} $$
⑤

Add $ \dfrac{248x^2+315x}{315} $ and $ \dfrac{x^2}{11} $ to get $ \dfrac{ \color{purple}{ 3043x^2+3465x } }{ 3465 }$.

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}{ 11 }$ and the second by $\color{blue}{ 315 }$.

$$ \begin{aligned} \frac{248x^2+315x}{315} + \frac{x^2}{11} & = \frac{ \left( 248x^2+315x \right) \cdot \color{blue}{ 11 }}{ 315 \cdot \color{blue}{ 11 }} + \frac{ x^2 \cdot \color{blue}{ 315 }}{ 11 \cdot \color{blue}{ 315 }} = \\[1ex] &=\frac{ \color{purple}{ 2728x^2+3465x } }{ 3465 } + \frac{ \color{purple}{ 315x^2 } }{ 3465 }=\frac{ \color{purple}{ 3043x^2+3465x } }{ 3465 } \end{aligned} $$
⑥

Add $ \dfrac{3043x^2+3465x}{3465} $ and $ \dfrac{x^2}{13} $ to get $ \dfrac{ \color{purple}{ 43024x^2+45045x } }{ 45045 }$.

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}{ 13 }$ and the second by $\color{blue}{ 3465 }$.

$$ \begin{aligned} \frac{3043x^2+3465x}{3465} + \frac{x^2}{13} & = \frac{ \left( 3043x^2+3465x \right) \cdot \color{blue}{ 13 }}{ 3465 \cdot \color{blue}{ 13 }} + \frac{ x^2 \cdot \color{blue}{ 3465 }}{ 13 \cdot \color{blue}{ 3465 }} = \\[1ex] &=\frac{ \color{purple}{ 39559x^2+45045x } }{ 45045 } + \frac{ \color{purple}{ 3465x^2 } }{ 45045 }=\frac{ \color{purple}{ 43024x^2+45045x } }{ 45045 } \end{aligned} $$
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