Add $ \dfrac{1}{2} $ and $ 4t^2 $ to get $ \dfrac{ \color{purple}{ 8t^2+1 } }{ 2 }$.

Step 1: Write $ 4t^2 $ 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 second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{1}{2} +4t^2 & \xlongequal{\text{Step 1}} \frac{1}{2} + \frac{4t^2}{\color{red}{1}} = \frac{ 1 }{ 2 } + \frac{ 4t^2 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 1 } }{ 2 } + \frac{ \color{purple}{ 8t^2 } }{ 2 }=\frac{ \color{purple}{ 8t^2+1 } }{ 2 } \end{aligned} $$

Add $ \dfrac{8t^2+1}{2} $ and $ \dfrac{1}{64t^2} $ to get $ \dfrac{ \color{purple}{ 512t^4+64t^2+2 } }{ 128t^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}{ 64t^2 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{8t^2+1}{2} + \frac{1}{64t^2} & = \frac{ \left( 8t^2+1 \right) \cdot \color{blue}{ 64t^2 }}{ 2 \cdot \color{blue}{ 64t^2 }} + \frac{ 1 \cdot \color{blue}{ 2 }}{ 64t^2 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 512t^4+64t^2 } }{ 128t^2 } + \frac{ \color{purple}{ 2 } }{ 128t^2 }=\frac{ \color{purple}{ 512t^4+64t^2+2 } }{ 128t^2 } \end{aligned} $$