Question

What is the square root of $0.0081 $?

Answer 1

Hint: In order to find the square root of $0.0081 $, first we need to change the decimal $0.0081 $into a fraction by looking at how many digits counting from right to left the decimal is placed. Now remove the decimal by dividing the value with power of $10 $ as the same number of digits counted after decimal. Then applying square root and solving by expanding the radicand in terms of its prime factors and then taking out the common values whose square roots are known.

Complete step-by-step answer:
We are given with the decimal $0.0081 $.
Since, we can see that the number of digits after the decimal is $4 $. So, divide only the real number value with no zero before decimal and no decimal point by $10 $ to the power of $4 $and we get:
 $
  0.0081 \\
   = \dfrac{{81}}{{{{10}^4}}} \;
  $
Expand $10 $ for $4 $, as we know that value can be written for the number of times the power is given and we get:
 $\dfrac{{81}}{{{{10}^4}}} = \dfrac{{81}}{{10000}} $
Applying square root to the value:
 $\sqrt {\dfrac{{81}}{{10000}}} $
Expanding the radicand in terms of their prime factors and we know that the factors of the values are as follows:
 $
  81 = 3 \times 3 \times 3 \times 3 \\
  10000 = 2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 5 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5 \;
  $
That implies:
\[\sqrt {\dfrac{{81}}{{10000}}} = \sqrt {\dfrac{{3 \times 3 \times 3 \times 3}}{{2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5}}} \]
Since, we know that two similar numbers present inside the root can be taken out as single unit, what we call its square root. For ex: $\sqrt {x \times x} = x $.
Similarly, we know that $\sqrt 9 = \sqrt {3 \times 3} = 3 $, $\sqrt 4 = \sqrt {2 \times 2} = 2 $, $\sqrt {25} = \sqrt {5 \times 5} = 5 $.
As we can see that there are $3 \times 3 \times 3 \times 3 $, so we can take out two 3’s from the root, similarly for $2 $ and $5 $, and we get: \[\sqrt {\dfrac{{3 \times 3 \times 3 \times 3}}{{2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5}}} = \dfrac{{3 \times 3}}{{2 \times 2 \times 5 \times 5}} = \dfrac{9}{{100}}\]
Since, there are two zeroes in the denominator, so placing a decimal after two digits counting from right and we get:
\[\dfrac{9}{{100}} = 0.09\].
Therefore, the square root of $0.0081 $ is $0.09 $.
So, the correct answer is “ $0.09 $”.

Note: We could have left the result in terms of fraction also if it’s not necessary to convert into decimal, it would not make any difference.
We can write the values as, \[81 = 9 \times 9\] and \[10000 = 10 \times 10 \times 10 \times 10\], instead writing it in terms of its prime factors.
Always cross check the answers once.