for what values of the variable does each of the following expressions make sense?

Question

sqaure root of 1+3a divided by 25

Answer 1

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:

1. The quantity inside the square root must be non-negative, because you cannot take the square root of a negative number in real numbers.
2. 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:

\[ a \geq -\frac{1}{3} \]