Question

1. \\(\frac{a^3 - 8a^2}{a^2 - 7a + 6}\\)

Explanation:

Step1: Factor numerator and denominator

Numerator: \(a^3 - 8a^2 = a^2(a - 8)\)
Denominator: \(a^2 - 7a + 6 = (a - 1)(a - 6)\) (by factoring quadratic: find two numbers with product 6 and sum -7, which are -1 and -6)

Step2: Simplify the fraction

The factored form is \(\frac{a^2(a - 8)}{(a - 1)(a - 6)}\). There are no common factors between numerator and denominator, so this is the simplified form (or we can note if we were to cancel, but here no common factors exist). Wait, wait—wait, did I factor denominator correctly? Let's re - check denominator: \(a^2-7a + 6\). The discriminant is \(49-24 = 25\), roots are \(\frac{7\pm5}{2}\), so \(a = 6\) and \(a = 1\), so \((a - 1)(a - 6)\) is correct. Numerator: \(a^3-8a^2=a^2(a - 8)\). So no common factors, so the simplified rational expression is \(\frac{a^2(a - 8)}{(a - 1)(a - 6)}\) or expanded numerator \(\frac{a^3 - 8a^2}{(a - 1)(a - 6)}\), but usually factored form is preferred.

Wait, maybe the original problem was to simplify, so the key steps are factoring numerator and denominator.

Answer:

\(\frac{a^2(a - 8)}{(a - 1)(a - 6)}\) (or \(\frac{a^3 - 8a^2}{a^2 - 7a + 6}\) if no simplification is possible, but factoring shows no common factors)