Subtract $ \dfrac{a^5}{3} $ from $ \dfrac{a^5}{6} $ to get $ \dfrac{ \color{purple}{ -a^5 } }{ 6 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{a^5}{6} - \frac{a^5}{3} & = \frac{ a^5 }{ 6 } - \frac{ a^5 \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ a^5 } }{ 6 } - \frac{ \color{purple}{ 2a^5 } }{ 6 }=\frac{ \color{purple}{ -a^5 } }{ 6 } \end{aligned} $$

Multiply $5$ by $ \dfrac{a^5}{12} $ to get $ \dfrac{ 5a^5 }{ 12 } $.

Step 1: Write $ 5 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 5 \cdot \frac{a^5}{12} & \xlongequal{\text{Step 1}} \frac{5}{\color{red}{1}} \cdot \frac{a^5}{12} \xlongequal{\text{Step 2}} \frac{ 5 \cdot a^5 }{ 1 \cdot 12 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5a^5 }{ 12 } \end{aligned} $$

Multiply $2$ by $ \dfrac{a^5}{15} $ to get $ \dfrac{ 2a^5 }{ 15 } $.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{a^5}{15} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{a^5}{15} \xlongequal{\text{Step 2}} \frac{ 2 \cdot a^5 }{ 1 \cdot 15 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2a^5 }{ 15 } \end{aligned} $$

Add $ \dfrac{-a^5}{6} $ and $ \dfrac{a^5}{2} $ to get $ \dfrac{ \color{purple}{ 2a^5 } }{ 6 }$.

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

$$ \begin{aligned} \frac{-a^5}{6} + \frac{a^5}{2} & = \frac{ -a^5 }{ 6 } + \frac{ a^5 \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ -a^5 } }{ 6 } + \frac{ \color{purple}{ 3a^5 } }{ 6 }=\frac{ \color{purple}{ 2a^5 } }{ 6 } \end{aligned} $$

Multiply $5$ by $ \dfrac{a^5}{12} $ to get $ \dfrac{ 5a^5 }{ 12 } $.

Step 1: Write $ 5 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 5 \cdot \frac{a^5}{12} & \xlongequal{\text{Step 1}} \frac{5}{\color{red}{1}} \cdot \frac{a^5}{12} \xlongequal{\text{Step 2}} \frac{ 5 \cdot a^5 }{ 1 \cdot 12 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5a^5 }{ 12 } \end{aligned} $$

Multiply $2$ by $ \dfrac{a^5}{15} $ to get $ \dfrac{ 2a^5 }{ 15 } $.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{a^5}{15} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{a^5}{15} \xlongequal{\text{Step 2}} \frac{ 2 \cdot a^5 }{ 1 \cdot 15 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2a^5 }{ 15 } \end{aligned} $$

Subtract $ \dfrac{5a^5}{12} $ from $ \dfrac{2a^5}{6} $ to get $ \dfrac{ \color{purple}{ -a^5 } }{ 12 }$.

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

$$ \begin{aligned} \frac{2a^5}{6} - \frac{5a^5}{12} & = \frac{ 2a^5 \cdot \color{blue}{ 2 }}{ 6 \cdot \color{blue}{ 2 }} - \frac{ 5a^5 }{ 12 } = \\[1ex] &=\frac{ \color{purple}{ 4a^5 } }{ 12 } - \frac{ \color{purple}{ 5a^5 } }{ 12 }=\frac{ \color{purple}{ -a^5 } }{ 12 } \end{aligned} $$

Multiply $2$ by $ \dfrac{a^5}{15} $ to get $ \dfrac{ 2a^5 }{ 15 } $.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{a^5}{15} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{a^5}{15} \xlongequal{\text{Step 2}} \frac{ 2 \cdot a^5 }{ 1 \cdot 15 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2a^5 }{ 15 } \end{aligned} $$

Subtract $ \dfrac{2a^5}{15} $ from $ \dfrac{-a^5}{12} $ to get $ \dfrac{ \color{purple}{ -13a^5 } }{ 60 }$.

To subtract 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 $\color{blue}{ 4 }$.

$$ \begin{aligned} \frac{-a^5}{12} - \frac{2a^5}{15} & = \frac{ \left( -a^5 \right) \cdot \color{blue}{ 5 }}{ 12 \cdot \color{blue}{ 5 }} - \frac{ 2a^5 \cdot \color{blue}{ 4 }}{ 15 \cdot \color{blue}{ 4 }} = \\[1ex] &=\frac{ \color{purple}{ -5a^5 } }{ 60 } - \frac{ \color{purple}{ 8a^5 } }{ 60 }=\frac{ \color{purple}{ -13a^5 } }{ 60 } \end{aligned} $$