Question

multiple choice 1 point
which expression is equivalent to (dfrac{(x^2 - 5x + 6)^{-1}}{(x - 2)^{-2}} div dfrac{(x - 3)^{-1}}{(x - 2)^{-2}})?
(dfrac{1}{x - 3})
(dfrac{(x - 2)^3}{(x - 3)^2})
(dfrac{(x + 2^3)}{(x - 3)^2})
(dfrac{1}{x - 2})

Explanation:

Step1: Rewrite negative exponents

Recall $a^{-n}=\frac{1}{a^n}$, so:
$$\frac{(x^2-5x+6)^{-1}}{(x-2)^{-2}} \div \frac{(x-3)^{-1}}{(x-2)^{-2}} = \frac{\frac{1}{x^2-5x+6}}{\frac{1}{(x-2)^2}} \div \frac{\frac{1}{x-3}}{\frac{1}{(x-2)^2}}$$

Step2: Simplify division of fractions

Dividing by a fraction is multiplying by its reciprocal:
$$\frac{(x-2)^2}{x^2-5x+6} \times \frac{(x-2)^2}{x-3}$$

Step3: Factor quadratic numerator

Factor $x^2-5x+6=(x-2)(x-3)$:
$$\frac{(x-2)^2}{(x-2)(x-3)} \times \frac{(x-2)^2}{x-3}$$

Step4: Cancel common terms

Cancel $(x-2)$ in first fraction:
$$\frac{x-2}{x-3} \times \frac{(x-2)^2}{x-3}$$

Step5: Multiply remaining fractions

Multiply numerators and denominators:
$$\frac{(x-2) \times (x-2)^2}{(x-3) \times (x-3)} = \frac{(x-2)^3}{(x-3)^2}$$

Answer:

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