Question

Find the value to three places of decimals of the following. It is given that

${\displaystyle \sqrt{2}=1.414}$, ${\displaystyle \sqrt{3}=1.732}$, ${\displaystyle \sqrt{5}=2.236}$ and ${\displaystyle \sqrt{10}=3.162}$

${\displaystyle \frac{\sqrt{2}-1}{\sqrt{5}}}$

Answer 1

We know that rationalization factor of the denominator is ${\displaystyle \sqrt{5}}$. We will multiply numerator and denominator of the given expression ${\displaystyle \frac{\sqrt{2}-1}{\sqrt{5}}}$by ${\displaystyle \sqrt{5}}$ to get

${\displaystyle \frac{\sqrt{2}-1}{\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{2}\times \sqrt{5}-\sqrt{5}}{\sqrt{5}\times \sqrt{5}}}$

${\displaystyle =\frac{\sqrt{10}-\sqrt{5}}{5}}$

Putting the value of ${\displaystyle \sqrt{10}}$and ${\displaystyle \sqrt{5}}$ we get

${\displaystyle \frac{\sqrt{10}-\sqrt{5}}{5}=\frac{3.162-2.236}{5}}$

${\displaystyle =\frac{0.926}{5}}$

= 0.1852

The value of expression 0.1852 can be round off tp three decimal places as 0.185.

given expression is simplified to 0.185