Answer:
[tex]=\frac{a+5-4\sqrt{a+1}}{a-3}[/tex]
Step-by-step explanation:
Given:
[tex]\frac{\sqrt{a+1}-2}{\sqrt{a+1}+2}[/tex]
Rationalise the denominator.
Solution:
Simplify the expression.
[tex]=\frac{\sqrt{a+1} -2}{\sqrt{a+1} +2 }[/tex]
Multiply numerator by both denominator and numerator.
[tex]=\frac{\sqrt{a+1} -2}{\sqrt{a+1}+2}\times \frac{\sqrt{a+1}-2}{\sqrt{a+1}-2}[/tex]
[tex]=\frac{(\sqrt{a+1}-2)(\sqrt{a+1}-2)}{(\sqrt{a+1}+2)(\sqrt{a+1}-2)}[/tex]
Assume [tex]\sqrt{a+1} =a\ and\ 2=b[/tex]
Applying formula [tex](a-b)(a+b)=a^{2} -b^{2}[/tex], so we get
[tex]=\frac{(\sqrt{a+1} -2)^{2}}{(\sqrt{a+1})^{2} -2^{2})}[/tex]
[tex]=\frac{(\sqrt{a+1})^{2} +2^{2}-2(\sqrt{a+1})(2)}{(a+1)-4}[/tex]
[tex]=\frac{(a+1) +4-4\sqrt{a+1}}{a+1-4}[/tex]
[tex]=\frac{a+5-4\sqrt{a+1}}{a-3}[/tex]
Therefore, the simplification of the expression is given below.
[tex]=\frac{a+5-4\sqrt{a+1}}{a-3}[/tex]
