Q48. - Algebra 2

Question

Simplify the expression $\frac{{\left(a^2-5a+6\right)}^{-1}}{{\left(a-2\right)}^{-2}}÷\frac{{\left(a-3\right)}^{-1}}{{\left(a-2\right)}^{-2}}$

Step-by-Step Solution

Answer

The simplified value of expression is $\frac{1}{a-2}$.

Step 1. Write down the given expression.

The given expression is $\frac{{\left(a^2-5a+6\right)}^{-1}}{{\left(a-2\right)}^{-2}}÷\frac{{\left(a-3\right)}^{-1}}{{\left(a-2\right)}^{-2}}$

Step 2. Simplify the expression.

The given expression $\frac{{\left(a^2-5a+6\right)}^{-1}}{{\left(a-2\right)}^{-2}}÷\frac{{\left(a-3\right)}^{-1}}{{\left(a-2\right)}^{-2}}$is simplified as:

$\begin{array}{c}\frac{{\left(a^2-5a+6\right)}^{-1}}{{\left(a-2\right)}^{-2}}÷\frac{{\left(a-3\right)}^{-1}}{{\left(a-2\right)}^{-2}}=\frac{{\left(a^2-5a+6\right)}^{-1}}{{\left(a-2\right)}^{-2}}\times \frac{{\left(a-2\right)}^{-2}}{{\left(a-3\right)}^{-1}}\\ =\frac{{\left(a-2\right)}^2}{\left(a^2-5a+6\right)}\times \frac{\left(a-3\right)}{{\left(a-2\right)}^2}\\ =\frac{a-3}{a^2-5a+6}\\ =\frac{a-3}{\left(a-2\right)\left(a-3\right)}\\ =\frac{1}{a-2}\end{array}$

Step 3. Conclusion.

The simplified value of expression is $\frac{1}{a-2}$.