Question

If $\sqrt{2}=1.414$, then $\sqrt{\frac{\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)}}$=?
(a) 0.207
(b) 2.414
(c) 0.414
(d) 0.621

Answer 1

(c) 0.414

On rationalising $\sqrt{{\displaystyle \frac{\sqrt{2}-1}{\sqrt{2}+1}}}$ by multiplying it by $\left(\sqrt{2}-1\right)$ both in the numerator and denominator, we get:

$$\begin{gathered} \sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}\times \frac{\sqrt{2}-1}{\sqrt{2}-1}} \\ =\sqrt{{\displaystyle \frac{{\left(\sqrt{2}-1\right)}^2}{{\left(\sqrt{2}\right)}^2-{\left(1\right)}^2}}} \\ =\frac{\sqrt{2}-1}{2-1} \\ =\sqrt{2}-1 \\ \because \sqrt{2}=1.414 \\ \therefore \sqrt{2}-1=1.414-1=0.414 \end{gathered}$$