Question

If $a={(\sqrt{5}+\sqrt{4})}^{-3}andb={(\sqrt{5}-\sqrt{4})}^{-3}$, then the value of ${(a+1)}^{-1}+{(b+1)}^{-1}$ is

• A. $20\sqrt{5}$
• B. 4
• C. 1
• D. $16\sqrt{5}$

Answer 1

The correct option is C 1

Given that

$a={(\sqrt{5}+\sqrt{4})}^{-3}\&b={(\sqrt{5}-\sqrt{4})}^{-3}.\therefore ab={(\sqrt{5}+\sqrt{4})}^{-3}\times {(\sqrt{5}-\sqrt{4})}^{-3}={(5-4)}^{-3}=1.\therefore {(a+1)}^{-1}+{(b+1)}^{-1}=\frac{1}{a+1}+\frac{1}{b+1}=\frac{a+b+2}{ab+a+b+1}.$\\

but ab=1.

So $\frac{a+b+2}{ab+a+b+1}$=$\frac{a+b+2}{a+b+2}$ =1.

Ans- Option C.