You cannot simply substitute $\sqrt{\frac{mg}{b}}$ with $\frac{mg}{b}$. Here's why:

• $\sqrt{\frac{mg}{b}}$ represents the square root of the ratio $\frac{mg}{b}$, which is dimensionally different from $\frac{mg}{b}$.
• $\frac{mg}{b}$ has the units of force divided by damping (if $b$ is a damping constant), while $\sqrt{\frac{mg}{b}}$ would have units related to velocity (or some other dimension depending on what $m$, $g$, and $b$ represent).

Thus, they cannot be directly substituted for each other without altering the meaning or the dimensions of the expression.

Would you like to explore simplifications or substitutions based on another part of the expression, or more details on dimensional analysis?

5 related questions:

1. What physical context does this equation represent (e.g., motion, energy)?
2. Can the exponential terms be simplified under certain assumptions, such as $t \rightarrow \infty$?
3. Is there a reason you want to remove the square root term?
4. How does the behavior of the expression change with respect to time $t$?
5. Would the substitution of $mg/b$ hold under any specific limit or approximation?

Tip:

When simplifying equations, always check the dimensional consistency before making substitutions, as this ensures physical correctness.