Question

simplify each expression.
\\(\frac{2}{n + 7} \div \frac{n^2 + 2n - 8}{n - 2}\\)
numerator/denominator = \\(\square\\)/\\(\square\\)

Explanation:

Step1: Recall division of fractions rule

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, \(\frac{2}{n + 7}\div\frac{n^{2}+2n - 8}{n - 2}=\frac{2}{n + 7}\times\frac{n - 2}{n^{2}+2n - 8}\)

Step2: Factor the quadratic in the denominator

Factor \(n^{2}+2n - 8\). We need two numbers that multiply to \(- 8\) and add to \(2\). Those numbers are \(4\) and \(-2\). So, \(n^{2}+2n - 8=(n + 4)(n - 2)\)

Step3: Substitute the factored form and simplify

Substitute \(n^{2}+2n - 8=(n + 4)(n - 2)\) into the expression: \(\frac{2}{n + 7}\times\frac{n - 2}{(n + 4)(n - 2)}\). The \((n - 2)\) terms cancel out (assuming \(n
eq2\) to avoid division by zero), giving \(\frac{2}{(n + 7)(n + 4)}\) or \(\frac{2}{n^{2}+11n + 28}\)

Answer:

\(2\) / \((n^{2}+11n + 28)\) (or \(2\) / \(( (n + 7)(n + 4) )\))