Given:
The expression is
[tex]\dfrac{2}{a-2}-\dfrac{8}{a^2-4}[/tex]
To find:
The simplified form of the given expression.
Solution:
We have,
[tex]\dfrac{2}{a-2}-\dfrac{8}{a^2-4}[/tex]
It can be written as
[tex]=\dfrac{2}{a-2}-\dfrac{8}{a^2-2^2}[/tex]
[tex]=\dfrac{2}{a-2}-\dfrac{8}{(a-2)(a+2)}[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
Taking LCM, we get
[tex]=\dfrac{2(a+2)-8}{(a-2)(a+2)}[/tex]
[tex]=\dfrac{2a+4-8}{(a-2)(a+2)}[/tex]
[tex]=\dfrac{2a-4}{(a-2)(a+2)}[/tex]
[tex]=\dfrac{2(a-2)}{(a-2)(a+2)}[/tex]
Cancel out the common factors.
[tex]=\dfrac{2}{a+2}[/tex]
Therefore, the simplified form of the given expression is [tex]\dfrac{2}{a+2}[/tex].
