To solve this problem, let's follow the steps to find the simplest form of the expression given:
Jesse is traveling up and down a stream in a kayak at an average rate of 5 miles/hour. The total distance of the round trip is 16 miles, which means each leg of the trip (upstream and downstream) is 8 miles. So, [tex]\( B = 8 \)[/tex].
The expression for the difference in time taken for upstream and downstream travel is given by:
[tex]\[ \frac{B}{5 - c} - \frac{B}{5 + c} \][/tex]
Substituting [tex]\( B = 8 \)[/tex], we get:
[tex]\[ \frac{8}{5 - c} - \frac{8}{5 + c} \][/tex]
To combine these fractions, we can find a common denominator. The common denominator for [tex]\( 5 - c \)[/tex] and [tex]\( 5 + c \)[/tex] is [tex]\( (5 - c)(5 + c) \)[/tex].
Thus, we write:
[tex]\[ \frac{8(5 + c)}{(5 - c)(5 + c)} - \frac{8(5 - c)}{(5 - c)(5 + c)} \][/tex]
Now, combine the numerators over the common denominator:
[tex]\[ \frac{8(5 + c) - 8(5 - c)}{(5 - c)(5 + c)} \][/tex]
Simplify the numerator:
[tex]\[ 8(5 + c) - 8(5 - c) = 40 + 8c - 40 + 8c = 16c \][/tex]
So the expression becomes:
[tex]\[ \frac{16c}{(5 - c)(5 + c)} \][/tex]
Notice that [tex]\( (5 - c)(5 + c) \)[/tex] is a difference of squares, which simplifies to:
[tex]\[ 25 - c^2 \][/tex]
Therefore, the simplified form of the expression is:
[tex]\[ \frac{16c}{25 - c^2} \][/tex]
Hence, the difference in time taken for upstream and downstream travel in simplest form is:
[tex]\[ \boxed{\frac{16c}{c^2 - 25}} \][/tex]
