Question

How do you simplify the given square root $\sqrt{\dfrac{1}{5}}$?

Answer 1

Hint: We start solving the problem by writing the given square root $\sqrt{\dfrac{1}{5}}$ equating to a variable. We then find the value of $\dfrac{1}{5}$ inside the square root to proceed through the variable. We then write the obtained result of division as a multiplication of two numbers to proceed further through the problem. We then make use of the fact that \[\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\] to proceed through the problem to get the simplified form of the given square root.

Complete step by step answer:
According to the problem, we are asked to simplify the given square root $\sqrt{\dfrac{1}{5}}$.
Let us assume $s=\sqrt{\dfrac{1}{5}}$.
$\Rightarrow s=\sqrt{0.2}$.
$\Rightarrow s=\sqrt{100\times 0.002}$ ---(1).
We can see that the equation (1) resembles the form $\sqrt{a\times b}$. We know that \[\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\]. Let us use this result in equation (1).
$\Rightarrow s=\sqrt{100}\times \sqrt{0.002}$ ---(2).
We know that $\sqrt{100}=10$. Let us use this result in equation (2).
$\Rightarrow s=10\times \sqrt{0.002}$.

$\therefore $ We have found the simplified form of the square root of $\sqrt{\dfrac{1}{5}}$ as $10\times \sqrt{0.002}$.

Note: Whenever we get this type of problems, we try of make use of the fact \[\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\] to make the process simpler. We can also simplify the answer further but we cannot get the exact answer for the square root. We can also find the estimate of the square root by making use of the long division method. Similarly, we can expect problems to find the square root of the number 1000 using a long division method up to 3 decimals.