Question

divide.
enter your answer as a single completely simplified rational expression with nonnegative exponents.
\\(\frac{n + 7}{5 - n} \div \frac{n^2 - 49}{n - 5}\\)

Explanation:

Step1: Rewrite division as multiplication

To divide by a fraction, multiply by its reciprocal. So, \(\frac{n + 7}{5 - n}\div\frac{n^{2}-49}{n - 5}=\frac{n + 7}{5 - n}\times\frac{n - 5}{n^{2}-49}\)

Step2: Factor the expressions

Factor \(n^{2}-49\) using the difference of squares: \(n^{2}-49=(n + 7)(n - 7)\). Also, note that \(5 - n=-(n - 5)\)
So the expression becomes \(\frac{n + 7}{-(n - 5)}\times\frac{n - 5}{(n + 7)(n - 7)}\)

Step3: Cancel common factors

Cancel out the common factors \((n + 7)\) and \((n - 5)\) (assuming \(n
eq5\) and \(n
eq - 7\) to avoid division by zero)
After canceling, we have \(\frac{1}{-(n - 7)}\)

Step4: Simplify the sign

\(\frac{1}{-(n - 7)}=\frac{1}{-n + 7}=\frac{1}{7 - n}\)

Answer:

\(\frac{1}{7 - n}\)