Question

Find the square root of $0.017424$ ?

Answer 1

Hint: At first, we convert the decimal $0.017424$ to fraction $\dfrac{17424}{1000000}$ . Now, we take the square root of the numerator and the denominator separately. We prime factorise the numerator and get $17424=\left( 2\times 2\times 2\times 2 \right)\times \left( 3\times 3 \right)\times \left( 11\times 11 \right)$ . After square rooting it becomes $132$ and the denominator becomes $1000$ . The entire number becomes $0.132$ .

Complete step by step answer:
The given decimal that we have at our disposal is,
$0.017424$
Decimals are just another way to represent fractions. Instead of writing two numbers above and below a bar, we sometimes prefer to write them as a single number with a decimal point somewhere in between the digits. Decimals are defined by division by $10$ and its multiples. So, $0.1$ means $\dfrac{1}{10}$ , $0.01$ means $\dfrac{1}{100}$ , $0.001$ means $\dfrac{1}{1000}$ and so on.
Decimal to fraction and fraction to decimals conversions are pretty easy. For fractions, if there is $10$ or any of its multiples in the denominator, then conversion to decimal can be done very easily. For this, at first, we will count the number of decimal places or the number of digits after the decimal point. We then multiply the number by $\dfrac{1}{{{10}^{n}}}$ where n is the number of decimal places and remove the decimal point.
In our case, $n=6$ . So, the decimal becomes,
$0.017424=17424\times \dfrac{1}{{{10}^{6}}}=17424\times \dfrac{1}{1000000}=\dfrac{17424}{1000000}$
Taking the square root of a fraction would mean taking the square roots of its numerator and denominator. But for square rooting, we need to prime factorise it.
$\begin{align}
  & 2\left| \!{\underline {\,
  17424 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8712 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4356 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2178 \,}} \right. \\
 & 3\left| \!{\underline {\,
  1089 \,}} \right. \\
 & 3\left| \!{\underline {\,
  363 \,}} \right. \\
 & 11\left| \!{\underline {\,
  121 \,}} \right. \\
 & ~~~~11 \\
\end{align}$
This gives $17424=\left( 2\times 2\times 2\times 2 \right)\times \left( 3\times 3 \right)\times \left( 11\times 11 \right)$ . Square rooting will give,
$\sqrt{17424}=\left( 2\times 2 \right)\times \left( 3 \right)\times \left( 11 \right)=132$
The square root of $1000000$ is clearly $1000$ . The resultant will be $\dfrac{132}{1000}=0.132$ .

Therefore, we can conclude that the square root of $0.017424$ will be $0.132$ .

Note: The decimal to fraction conversion should be dealt with care, especially while counting the number of decimal places. This problem can be solved in another way. We can express $0.017424$ as ${{2}^{4}}\times {{3}^{2}}\times {{11}^{2}}\times {{10}^{-6}}$ which can be easily square rooted.