Question

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\\\frac{n - 7}{(n - 7)(n - 2)} \cdot \frac{6n^2(2 - n)}{2n^2}\\

Explanation:

Identify the given expression

We are given the following product of rational expressions:

$$ \frac{n - 7}{(n - 7)(n - 2)} \cdot \frac{6n^2(2 - n)}{2n^2} $$

Simplify the first rational expression

We can cancel the common factor \(n - 7\) in the numerator and denominator of the first fraction, assuming \(n
eq 7\):

$$ \frac{1}{n - 2} \cdot \frac{6n^2(2 - n)}{2n^2} $$

Simplify the second rational expression

We can simplify the coefficients and the variable terms in the second fraction:

$$ \frac{6n^2}{2n^2} = 3 \quad (n eq 0) $$

This simplifies the product to:

$$ \frac{1}{n - 2} \cdot 3(2 - n) $$

Relate the remaining binomial factors

Notice that \(2 - n\) is the opposite of \(n - 2\):

$$ 2 - n = -(n - 2) $$

Substitute this back into the expression:

$$ \frac{3 \cdot [-(n - 2)]}{n - 2} $$

Perform final cancellation

Cancel the common factor \(n - 2\), assuming \(n
eq 2\):

$$ 3 \cdot (-1) = -3 $$

Answer:

\(-3\)