To determine the values of the variable \(a\) for which the expression \(\frac{\sqrt{1 + 3a}}{25}\) makes sense, we need to consider two main conditions:
- The quantity inside the square root must be non-negative, because you cannot take the square root of a negative number in real numbers.
- The denominator must not be zero, although in this case, the denominator is a constant \(25\), which is never zero.
Let's focus on the first condition:
\[ 1 + 3a \geq 0 \]
To solve for \(a\), we can rearrange this inequality:
\[ 3a \geq -1 \]
Now, dividing both sides by \(3\):
\[ a \geq -\frac{1}{3} \]
So, the expression \(\frac{\sqrt{1 + 3a}}{25}\) makes sense for:
\[ a \geq -\frac{1}{3} \]