Sure, let's break down the given mathematical expression step by step and simplify it.
We start with the given expression:
[tex]\[
\frac{8}{5 - c} - \frac{8}{5 + c}
\][/tex]
To simplify this, we should first get a common denominator for the two fractions. The denominators in this case are [tex]\(5 - c\)[/tex] and [tex]\(5 + c\)[/tex]. The common denominator would be the product of these two denominators, which is [tex]\((5 - c)(5 + c)\)[/tex].
Rewriting each fraction with the common denominator, we get:
[tex]\[
\frac{8(5 + c)}{(5 - c)(5 + c)} - \frac{8(5 - c)}{(5 - c)(5 + c)}
\][/tex]
Now, we can combine these two fractions into a single fraction:
[tex]\[
\frac{8(5 + c) - 8(5 - c)}{(5 - c)(5 + c)}
\][/tex]
Next, we simplify the numerator:
[tex]\[
8(5 + c) - 8(5 - c) = 8 \cdot 5 + 8 \cdot c - 8 \cdot 5 + 8 \cdot c = 8c + 8c = 16c
\][/tex]
So, the simplified form of the numerator is [tex]\(16c\)[/tex].
The denominator, [tex]\((5 - c)(5 + c)\)[/tex], is a difference of squares and can be simplified to:
[tex]\[
5^2 - c^2 = 25 - c^2
\][/tex]
Putting it all together, we can now write the simplified fraction as:
[tex]\[
\frac{16c}{25 - c^2}
\][/tex]
Therefore, the simplified expression for the difference in time it takes Jesse to travel upstream and downstream is:
[tex]\[
\boxed{-\frac{16c}{c^2 - 25}}
\][/tex]
