Find the square root of 2.0164.

Hint: Write 2.0164 in the form of $\dfrac{p}{q}$, where p and q are integers and q is non-zero. Note that this can be done since 2.0164 is a rational number. Use the property \[\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}\]. Find the prime factorisation of p and q to find the values of $\sqrt{p}$ and $\sqrt{q}$, and hence the value of $\sqrt{\dfrac{p}{q}}$.

Complete step-by-step answer:
We know that $2.0164=\dfrac{20164}{10000}$
We need to write the above fraction in lowest terms. For that, we will calculate the gcd of 20164 and 10000.
We will use the long division method (Euclid’s division algorithm) to find the gcd of 20164 and 10000.

We have
\[\begin{align}
  & 20164\text{ }=\text{ }10000\times 2+164 \\
 & 10000=164\times 60+160 \\
 & 164=160\times 1+4 \\
 & 160=4\times 40+0 \\
\end{align}\]
Hence gcd (20164,10000) = 4.

Dividing numerator and denominator by gcd(20164,10000), we get
$2.0164=\dfrac{5041}{2500}$
Now we have
$\begin{align}
  & 5041={{71}^{2}} \\
 & 2500={{5}^{4}}\times {{2}^{2}} \\
\end{align}$
Hence $\sqrt{5041}=71$ and $\sqrt{2500}={{5}^{2}}\times 2$

We know that \[\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}\].
Using the above result, we get
\[\sqrt{\dfrac{5041}{2500}}=\dfrac{\sqrt{5041}}{\sqrt{2500}}=\dfrac{71}{{{5}^{2}}\times 2}=\dfrac{71}{50}=1.42\]
Hence the square root of 2.0164 is equal to 1.42.

Note: Square root or the principal square root of a number x is a number y such that ${{y}^{2}}=x,y>0$.
Verification of 1.42 is a square root of 2.0164: Clearly 1.42 >0
Also, we have \[{{142}^{2}}=20164\]
Hence \[{{1.42}^{2}}=2.0164\]
Hence 1.42 is the square of 2.0164.