Step 1
Given:
[tex](x+2)\times\frac{(x-2)}{(x+2)_{}}\div\frac{x^2-4}{x-6}[/tex]
Required: To simplify the problem.
Step 2
Apply PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
As a rule, we will move from left to right. This means we will multiply first before dividing.
Therefore for the first part
[tex](x+2)\times\frac{(x-2)}{(x+2)_{}}=\text{ x-2}[/tex]
Hence we can write,
[tex](x-2)\div\frac{x^2-4}{x-6}[/tex]
Step 3
Expand x²-4 using the difference of two squares.
[tex]\begin{gathered} (a+b)^2=(a+b)(a-b) \\ \text{Hence,} \\ x^2-4=(x^2-2^2) \\ x^2-4=(x+2)(x-2) \end{gathered}[/tex]
Hence, we can write
[tex](x-2)\div\frac{(x+2)(x-2)_{}}{x-6}[/tex]
Step 4
Simplify step 3
[tex]\begin{gathered} =(x-2)\times\frac{x-6}{(x+2)(x-2)} \\ =\frac{x-6}{x+2} \end{gathered}[/tex]
Hence after simplification, the answer is
[tex]\frac{x-6}{x+2}[/tex]
