Simplify (((x^2-5x+6)^-1)/((x-2)^-2))÷(((x-3)^-1)/((x-2)^-2))

$\frac{{(x^{2} - 5 x + 6)}^{- 1}}{{(x - 2)}^{- 2}} \div \frac{{(x - 3)}^{- 1}}{{(x - 2)}^{- 2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{{(x^{2} - 5 x + 6)}^{- 1}}{{(x - 2)}^{- 2}} \cdot \frac{{(x - 2)}^{- 2}}{{(x - 3)}^{- 1}}$

Step 2

Simplify terms.

Combine.

$\frac{{(x^{2} - 5 x + 6)}^{- 1} {(x - 2)}^{- 2}}{{(x - 2)}^{- 2} {(x - 3)}^{- 1}}$

Raise to zero.

Move ${(x^{2} - 5 x + 6)}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(x - 2)}^{- 2}}{{(x - 2)}^{- 2} {(x - 3)}^{- 1} (x^{2} - 5 x + 6)}$

Move ${(x - 3)}^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(x - 2)}^{- 2} (x - 3)}{{(x - 2)}^{- 2} (x^{2} - 5 x + 6)}$

Cancel the common factor of ${(x - 2)}^{- 2}$ .

Cancel the common factor.

$\frac{{(x - 2)}^{- 2} (x - 3)}{{(x - 2)}^{- 2} (x^{2} - 5 x + 6)}$

Rewrite the expression.

$\frac{x - 3}{x^{2} - 5 x + 6}$

Step 3

Factor $x^{2} - 5 x + 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

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

Step 4

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (((x^2-5x+6)^-1)/((x-2)^-2))÷(((x-3)^-1)/((x-2)^-2))

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