Simplify (x-2)/(3x^3)*(9x^2)/(x^2-4x+4)

$\frac{x - 2}{3 x^{3}} \cdot \frac{9 x^{2}}{x^{2} - 4 x + 4}$

Step 1

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

$\frac{x - 2}{3 x^{3}} \cdot \frac{9 x^{2}}{x^{2} - 4 x + 2^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

$\frac{x - 2}{3 x^{3}} \cdot \frac{9 x^{2}}{x^{2} - 2 \cdot x \cdot 2 + 2^{2}}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

$\frac{x - 2}{3 x^{3}} \cdot \frac{9 x^{2}}{{(x - 2)}^{2}}$

Step 2

Simplify terms.

Cancel the common factor of $x - 2$ .

Factor $x - 2$ out of ${(x - 2)}^{2}$ .

$\frac{x - 2}{3 x^{3}} \cdot \frac{9 x^{2}}{(x - 2) (x - 2)}$

Cancel the common factor.

$\frac{x - 2}{3 x^{3}} \cdot \frac{9 x^{2}}{(x - 2) (x - 2)}$

Rewrite the expression.

$\frac{1}{3 x^{3}} \cdot \frac{9 x^{2}}{x - 2}$

Cancel the common factor of $3 x^{2}$ .

Factor $3 x^{2}$ out of $3 x^{3}$ .

$\frac{1}{3 x^{2} (x)} \cdot \frac{9 x^{2}}{x - 2}$

Factor $3 x^{2}$ out of $9 x^{2}$ .

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

Cancel the common factor.

$\frac{1}{3 x^{2} x} \cdot \frac{3 x^{2} \cdot 3}{x - 2}$

Rewrite the expression.

$\frac{1}{x} \cdot \frac{3}{x - 2}$

Multiply $\frac{1}{x}$ by $\frac{3}{x - 2}$ .

$\frac{3}{x (x - 2)}$