$$x\cdot\dfrac{4}{3}r^3-y(\dfrac{4}{3}r^3-\dfrac{h^2}{3}(3r-h))=\dfrac{-h^3y+3h^2ry+4r^3x-4r^3y}{3}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x\cdot\frac{4}{3}r^3-y(\frac{4}{3}r^3-\frac{h^2}{3}(3r-h))& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4x}{3}r^3-y(\frac{4r^3}{3}-\frac{-h^3+3h^2r}{3}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{4r^3x}{3}-y\frac{h^3-3h^2r+4r^3}{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{4r^3x}{3}-\frac{h^3y-3h^2ry+4r^3y}{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{-h^3y+3h^2ry+4r^3x-4r^3y}{3}\end{aligned} $$ | |
| ① | Multiply $x$ by $ \dfrac{4}{3} $ to get $ \dfrac{ 4x }{ 3 } $. Step 1: Write $ x $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} x \cdot \frac{4}{3} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} \cdot \frac{4}{3} \xlongequal{\text{Step 2}} \frac{ x \cdot 4 }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ 4x }{ 3 } \end{aligned} $$ |
| ② | Multiply $ \dfrac{4}{3} $ by $ r^3 $ to get $ \dfrac{ 4r^3 }{ 3 } $. Step 1: Write $ r^3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{4}{3} \cdot r^3 & \xlongequal{\text{Step 1}} \frac{4}{3} \cdot \frac{r^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4 \cdot r^3 }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4r^3 }{ 3 } \end{aligned} $$ |
| ③ | Multiply $ \dfrac{h^2}{3} $ by $ 3r-h $ to get $ \dfrac{-h^3+3h^2r}{3} $. Step 1: Write $ 3r-h $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{h^2}{3} \cdot 3r-h & \xlongequal{\text{Step 1}} \frac{h^2}{3} \cdot \frac{3r-h}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ h^2 \cdot \left( 3r-h \right) }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3h^2r-h^3 }{ 3 } = \frac{-h^3+3h^2r}{3} \end{aligned} $$ |
| ④ | Multiply $ \dfrac{4x}{3} $ by $ r^3 $ to get $ \dfrac{ 4r^3x }{ 3 } $. Step 1: Write $ r^3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{4x}{3} \cdot r^3 & \xlongequal{\text{Step 1}} \frac{4x}{3} \cdot \frac{r^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4x \cdot r^3 }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4r^3x }{ 3 } \end{aligned} $$ |
| ⑤ | Subtract $ \dfrac{-h^3+3h^2r}{3} $ from $ \dfrac{4r^3}{3} $ to get $ \dfrac{h^3-3h^2r+4r^3}{3} $. To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator. $$ \begin{aligned} \frac{4r^3}{3} - \frac{-h^3+3h^2r}{3} & = \frac{4r^3}{\color{blue}{3}} - \frac{-h^3+3h^2r}{\color{blue}{3}} = \\[1ex] &=\frac{ 4r^3 - \left( -h^3+3h^2r \right) }{ \color{blue}{ 3 }}= \frac{h^3-3h^2r+4r^3}{3} \end{aligned} $$ |
| ⑥ | Multiply $ \dfrac{4x}{3} $ by $ r^3 $ to get $ \dfrac{ 4r^3x }{ 3 } $. Step 1: Write $ r^3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{4x}{3} \cdot r^3 & \xlongequal{\text{Step 1}} \frac{4x}{3} \cdot \frac{r^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4x \cdot r^3 }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4r^3x }{ 3 } \end{aligned} $$ |
| ⑦ | Multiply $y$ by $ \dfrac{h^3-3h^2r+4r^3}{3} $ to get $ \dfrac{ h^3y-3h^2ry+4r^3y }{ 3 } $. Step 1: Write $ y $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} y \cdot \frac{h^3-3h^2r+4r^3}{3} & \xlongequal{\text{Step 1}} \frac{y}{\color{red}{1}} \cdot \frac{h^3-3h^2r+4r^3}{3} \xlongequal{\text{Step 2}} \frac{ y \cdot \left( h^3-3h^2r+4r^3 \right) }{ 1 \cdot 3 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ h^3y-3h^2ry+4r^3y }{ 3 } \end{aligned} $$ |
| ⑧ | Subtract $ \dfrac{h^3y-3h^2ry+4r^3y}{3} $ from $ \dfrac{4r^3x}{3} $ to get $ \dfrac{-h^3y+3h^2ry+4r^3x-4r^3y}{3} $. To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator. $$ \begin{aligned} \frac{4r^3x}{3} - \frac{h^3y-3h^2ry+4r^3y}{3} & = \frac{4r^3x}{\color{blue}{3}} - \frac{h^3y-3h^2ry+4r^3y}{\color{blue}{3}} = \\[1ex] &=\frac{ 4r^3x - \left( h^3y-3h^2ry+4r^3y \right) }{ \color{blue}{ 3 }}= \frac{-h^3y+3h^2ry+4r^3x-4r^3y}{3} \end{aligned} $$ |