Rewrite \({a}^{2}-6a+9\) in the form \({a}^{2}-2ab+{b}^{2}\), where \(a=a\) and \(b=3\).
\[\frac{\frac{{a}^{2}-2(a)(3)+{3}^{2}}{{a}^{2}-2a-3}}{\frac{{a}^{2}-5a-6}{{a}^{2}-3a+2}}\times \frac{{(a-3)}^{2}}{(a-1)(a-2)}\]
Use Square of Difference: \({(a-b)}^{2}={a}^{2}-2ab+{b}^{2}\).
\[\frac{\frac{{(a-3)}^{2}}{{a}^{2}-2a-3}}{\frac{{a}^{2}-5a-6}{{a}^{2}-3a+2}}\times \frac{{(a-3)}^{2}}{(a-1)(a-2)}\]
Factor \({a}^{2}-2a-3\).
Ask: Which two numbers add up to \(-2\) and multiply to \(-3\)?
Rewrite the expression using the above.
\[(a-3)(a+1)\]
\[\frac{\frac{{(a-3)}^{2}}{(a-3)(a+1)}}{\frac{{a}^{2}-5a-6}{{a}^{2}-3a+2}}\times \frac{{(a-3)}^{2}}{(a-1)(a-2)}\]
Factor \({a}^{2}-5a-6\).
Ask: Which two numbers add up to \(-5\) and multiply to \(-6\)?
Rewrite the expression using the above.
\[(a-6)(a+1)\]
\[\frac{\frac{{(a-3)}^{2}}{(a-3)(a+1)}}{\frac{(a-6)(a+1)}{{a}^{2}-3a+2}}\times \frac{{(a-3)}^{2}}{(a-1)(a-2)}\]
Factor \({a}^{2}-3a+2\).
Ask: Which two numbers add up to \(-3\) and multiply to \(2\)?
Rewrite the expression using the above.
\[(a-2)(a-1)\]
\[\frac{\frac{{(a-3)}^{2}}{(a-3)(a+1)}}{\frac{(a-6)(a+1)}{(a-2)(a-1)}}\times \frac{{(a-3)}^{2}}{(a-1)(a-2)}\]
Invert and multiply.
\[\frac{{(a-3)}^{2}}{(a-3)(a+1)}\times \frac{(a-2)(a-1)}{(a-6)(a+1)}\times \frac{{(a-3)}^{2}}{(a-1)(a-2)}\]
Simplify \(\frac{{(a-3)}^{2}}{(a-3)(a+1)}\) to \(\frac{a-3}{a+1}\).
\[\frac{a-3}{a+1}\times \frac{(a-2)(a-1)}{(a-6)(a+1)}\times \frac{{(a-3)}^{2}}{(a-1)(a-2)}\]
Cancel \(a-1\).
\[\frac{a-3}{a+1}\times \frac{a-2}{(a-6)(a+1)}\times \frac{{(a-3)}^{2}}{a-2}\]
Cancel \(a-2\).
\[\frac{a-3}{a+1}\times \frac{1}{(a-6)(a+1)}{(a-3)}^{2}\]
Use this rule: \(\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\).
\[\frac{(a-3)\times 1\times {(a-3)}^{2}}{(a+1)(a-6)(a+1)}\]
Simplify \((a-3)\times 1\times {(a-3)}^{2}\) to \((a-3){(a-3)}^{2}\).
\[\frac{(a-3){(a-3)}^{2}}{(a+1)(a-6)(a+1)}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{{(a-3)}^{3}}{(a+1)(a-6)(a+1)}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{{(a-3)}^{3}}{{(a+1)}^{2}(a-6)}\]
(a-3)^3/((a+1)^2*(a-6))
