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Find the square root of $0.0081$

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Last updated date: 04th Mar 2024
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IVSAT 2024
Answer
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Hint: First we have to define what the terms we need to solve the problem are.
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 the right to left decimal are placed.
Now remove the decimal by dividing the value with power of $10$ as the same number of digits counted after decimal then apply square root and solve

Complete step-by-step solution:
From the given question, we have decimal $0.0081$.
Since digits after the decimal is $4$ numbers. So, divide only the real number value with no zero before decimal and no decimal point by $10$ to the power of $4$ thus we get
$0.0081$$ = \dfrac{{81}}{{{{10}^4}}}$ Expand \[10\] for \[4\] and the number of times in given $ \Rightarrow \dfrac{{81}}{{{{10}^4}}} = \dfrac{{81}}{{10000}}$
Now applying square root of the value, we get $\sqrt {\dfrac{{81}}{{10000}}} $
Let expanding the factors which is $81 = 3 \times 3 \times 3 \times 3$ and
$10000 = 2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 5$
Thus, applying the square root know as $\sqrt {y \times y} = y$
Hence by this we get $\sqrt {\dfrac{{81}}{{10000}}} $= $\sqrt {\dfrac{{3 \times 3 \times 3 \times 3}}{{2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 5 \times}}} = \dfrac{{3 \times 3}}{{2 \times 5 \times 2 \times 5}}$$ = \dfrac{9}{{100}}$
Since there are two zeros in the denominator, so placing a decimal after two digits counting from the right and we get
$ \Rightarrow \dfrac{9}{{100}} = 0.09$
Hence the square root of $0.0081$ will be \[0.09\]

Note: A square root of a number is a value that, when multiplied by itself, gives the number.
Example: \[4 \times 4 = 16\], so a square root of $16$ is $4$
We can able to simply solve this by applying square root on $0.0081$ which is $\sqrt {0.0081} = 0.09$
We can also write the values as, $81 = 9 \times 9$ and $10000 = 10 \times 10 \times 10 \times 10$, instead of replacing prime factor terms.
Square root of negative values will give imaginary axis; example; $\sqrt { - 1} = i$

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