Example

3(r+2s)=2t-4
a⋅(b-2)=3b
3(r+2s)=2t-4
a⋅(b-2)=3b

Question

$$\frac{ a-2 }{ { a }^{ 2 } +4a+4 } + \frac{ a+2 }{ a-2 }$$

Answer

$$((a-2)^2+(a+2)^3)/((a+2)^2*(a-2))$$

Solution


Rewrite \({a}^{2}+4a+4\) in the form \({a}^{2}+2ab+{b}^{2}\), where \(a=a\) and \(b=2\).
\[\frac{a-2}{{a}^{2}+2(a)(2)+{2}^{2}}+\frac{a+2}{a-2}\]
Use Square of Sum: \({(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\).
\[\frac{a-2}{{(a+2)}^{2}}+\frac{a+2}{a-2}\]
Rewrite the expression with a common denominator.
\[\frac{(a-2)(a-2)+(a+2){(a+2)}^{2}}{{(a+2)}^{2}(a-2)}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{{(a-2)}^{2}+(a+2){(a+2)}^{2}}{{(a+2)}^{2}(a-2)}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{{(a-2)}^{2}+{(a+2)}^{3}}{{(a+2)}^{2}(a-2)}\]
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