Question

question
fully simplify the expression below and write your answer as a single fraction.
$\frac{x(x - 4)^2}{x^2 - 5x + 4} cdot \frac{6x^2 - 60x + 54}{6x^3 - 78x^2 + 216x}$

Explanation:

Step1: Factor all numerators/denominators

First fraction:
Numerator: $x(x-4)^2$ (already factored)
Denominator: $x^2-5x+4=(x-1)(x-4)$
Second fraction:
Numerator: $6x^2-60x+54=6(x^2-10x+9)=6(x-1)(x-9)$
Denominator: $6x^3-78x^2+216x=6x(x^2-13x+36)=6x(x-4)(x-9)$

Step2: Rewrite the expression

$$\frac{x(x-4)^2}{(x-1)(x-4)} \cdot \frac{6(x-1)(x-9)}{6x(x-4)(x-9)}$$

Step3: Cancel common factors

Cancel $x$, $(x-4)$, $(x-1)$, $6$, $(x-9)$ from numerator and denominator:
$$\frac{(x-4)}{x(x-13)}$$

Answer:

$\frac{x-4}{x(x-13)}$