Question

Find the square root of the decimal fraction $0.000004$.

Answer 1

Hint: First we have to convert the given decimal into the fraction form. This can be done by multiplying and dividing it by 10 to the power of the number of digits after the decimal point. Then, we can find the square root by expanding it into two products of prime factors.

Complete step-by-step answer:
We have to find the square root of decimal fraction $0.000004$.
Here in $0.000004$ decimal is at sixth place, i.e there are six digits after the decimal point. So to convert $0.000004$ to fraction form it will be 4 in numerator and $1000000$.
Hence the fraction form of $0.000004$ is $\dfrac{4}{1000000}$.
We have to find the square root of $0.000004$ means we have to find the square root of $\dfrac{4}{1000000}$.
Square root of $\dfrac{4}{1000000}$ $=\sqrt{\dfrac{4}{1000000}}$.
Now we know that the product of prime factors of $4$ is $4=2\times 2$. Also the product of prime factors of $1000000$ is $1000000=2\times 2\times 2\times 2\times 2\times 2\times 5\times 5\times 5\times 5\times 5\times 5$.
So we get $\sqrt{\dfrac{4}{1000000}}=\sqrt{\dfrac{2\times 2}{2\times 2\times 2\times 2\times 2\times 2\times 5\times 5\times 5\times 5\times 5\times 5}}$.
Writing this in square form we get $\sqrt{\dfrac{4}{1000000}}=\sqrt{\dfrac{{{2}^{2}}}{{{2}^{2}}\times {{2}^{2}}\times {{2}^{2}}\times {{5}^{2}}\times {{5}^{2}}\times {{5}^{2}}}}$.
Rearranging the above expression we get \[\sqrt{\dfrac{4}{1000000}}=\sqrt{\dfrac{{{2}^{2}}}{{{\left( 2\times 2\times 2\times 5\times 5\times 5 \right)}^{2}}}}\].
Since we know that form any positive number $x$, $\sqrt{{{x}^{2}}}=x$.
Using this formula we \[\sqrt{\dfrac{{{2}^{2}}}{{{\left( 2\times 2\times 2\times 5\times 5\times 5 \right)}^{2}}}}=\dfrac{2}{\left( 2\times 2\times 2\times 5\times 5\times 5 \right)}\].
Simplifying we get \[\dfrac{2}{\left( 2\times 2\times 2\times 5\times 5\times 5 \right)}=\dfrac{2}{1000}\].
Therefore we have $\sqrt{\dfrac{4}{1000000}}=\dfrac{2}{1000}$.
In the fraction $\dfrac{2}{1000}$ there are three zeros in the denominator. The decimal form of $\dfrac{2}{1000}$ will be $0.002$.
So we get $\sqrt{\dfrac{4}{1000000}}=0.002$.
So we have $0.000004=0.002$.
Hence the square root of $0.000004$is $0.002$.
This is the required solution.

Note: In this problem the main key is converting a given decimal number to fraction number. So students must take care while converting the fraction and while converting the square root of fraction form to decimal form. Need to write the answer in decimal form not the simplified form of the fraction.