Question

Find the value of $\sqrt{\mathrm{0.444....}}$ in decimal form.

• A. 0.222.....
• B. 0.444.....
• C. 0.666....
• D. 0.888....

Answer 1

The correct option is C 0.666....

Given operation is $\sqrt{\mathrm{0.444....}}$
Now, $\mathrm{0.444....}=4\times \mathrm{0.111...}$............(1)
Let '$x$' be 0.111....
Multilpy $x$ with 10.
$\Rightarrow$ $10x$ = 1.11....
= 1 + 0.111....
= 1 + $x$ $(\because x=\mathrm{0.111....})$
$\Rightarrow$ $9x$ = 1
$\Rightarrow$ $x$ = $\frac{1}{9}$
$\therefore$ 0.111.... = $\frac{1}{9}$................(2)
Substituting the value of 0.111... in (1) we get,
$\mathrm{0.444....}=4\times \frac{1}{9}$
$=\frac{4}{9}$
$\therefore \sqrt{\mathrm{0.444....}}=\sqrt{\frac{4}{9}}$
$=\frac{2}{3}$
$=\left(\frac{2}{3}\right)\times \left(\frac{3}{3}\right)$
$=\frac{6}{9}$
$\because \frac{1}{9}=\mathrm{0.111....}$ [from eq (2)]
$\Rightarrow \sqrt{\mathrm{0.444....}}=6\times \mathrm{0.111.....}$
$=\mathrm{0.666......}$