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Sagot :
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]
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]
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