Question

The value of $\sqrt {\dfrac{{{{\left( {0.1} \right)}^2} + {{\left( {0.01} \right)}^2} + {{\left( {0.009} \right)}^2}}}{{{{\left( {0.01} \right)}^2} + {{\left( {0.001} \right)}^2} + {{\left( {0.0009} \right)}^2}}}} $

Answer 1

Hint: We need to think in a logical manner to solve these types of questions . First we need to change the terms in the denominator like ${\left( {0.001} \right)^2}$can be written as ${\left( {\dfrac{{0.01}}{{10}}} \right)^2}$ and then taking ${\left( {\dfrac{1}{{10}}} \right)^2}$common in the denominator we can see that the elements in the numerator and denominator is the same and hence cancelling and simplifying we get the required value .

Complete step-by-step answer:
We are asked to find the value of $\sqrt {\dfrac{{{{\left( {0.1} \right)}^2} + {{\left( {0.01} \right)}^2} + {{\left( {0.009} \right)}^2}}}{{{{\left( {0.01} \right)}^2} + {{\left( {0.001} \right)}^2} + {{\left( {0.0009} \right)}^2}}}} $
Now first lets deal with the denominator first
The first term of the denominator is ${\left( {0.01} \right)^2}$
This can be written as ${\left( {\dfrac{{0.1}}{{10}}} \right)^2}$
Same way lets write the other terms of the denominator
That is , ${\left( {0.001} \right)^2}$can be written as ${\left( {\dfrac{{0.01}}{{10}}} \right)^2}$
And , ${\left( {0.0009} \right)^2}$can be written as ${\left( {\dfrac{{0.009}}{{10}}} \right)^2}$
Therefore the given expression becomes
$ \Rightarrow \sqrt {\dfrac{{{{\left( {0.1} \right)}^2} + {{\left( {0.01} \right)}^2} + {{\left( {0.009} \right)}^2}}}{{{{\left( {\dfrac{{0.1}}{{10}}} \right)}^2} + {{\left( {\dfrac{{0.01}}{{10}}} \right)}^2} + {{\left( {\dfrac{{0.009}}{{10}}} \right)}^2}}}} $
Now lets take ${\left( {\dfrac{1}{{10}}} \right)^2}$ common in the denominator
$ \Rightarrow \sqrt {\dfrac{{{{\left( {0.1} \right)}^2} + {{\left( {0.01} \right)}^2} + {{\left( {0.009} \right)}^2}}}{{\dfrac{1}{{{{10}^2}}}\left[ {{{\left( {0.1} \right)}^2} + {{\left( {0.01} \right)}^2} + {{\left( {0.009} \right)}^2}} \right]}}} $
Simplifying we get
$
   \Rightarrow \sqrt {\dfrac{1}{{\dfrac{1}{{{{10}^2}}}}}} \\
   \Rightarrow \sqrt {{{10}^2}} \\
   \Rightarrow 10 \\
$
Hence we get the value of $\sqrt {\dfrac{{{{\left( {0.1} \right)}^2} + {{\left( {0.01} \right)}^2} + {{\left( {0.009} \right)}^2}}}{{{{\left( {0.01} \right)}^2} + {{\left( {0.001} \right)}^2} + {{\left( {0.0009} \right)}^2}}}} $ to be 10.

Additional information :
Decimals are based on the preceding powers of 10. Thus, as we move from left to right, the place value of digits gets divided by 10, meaning the decimal place value determines the tenths, hundredths and thousandths. A tenth means one tenth or $\dfrac{1}{{10}}$ . In decimal form, it is $0.1$ .

Note: Many students end up calculating the square value of the given numbers and simplifying it using BODMAS but it makes the process tedious and the error rate is high.