Question

What is the square root of $\dfrac{1}{4}$ in simplified radical form?
(a) 1
(b) $\dfrac{1}{4}$
(c) $\dfrac{1}{2}$
(d) None of these

Answer 1

Hint: We are trying to find the square root of $\dfrac{1}{4}$ in simplified radical form. So, start with, we will find the square roots of the denominator and numerator differently. Then, if we eliminate the negative value, we are getting the needed solution in a simplified given form.

Complete step by step solution:
According to the problem, we are trying to find the value of the square root of $\dfrac{1}{4}$in simplified radical form.
Now, expressing in simplest radical form just means simplifying a radical so that there are no more square roots, cube roots, 4th roots left to find. It also means removing any radicals in the denominator of a fraction.
Now, we have to get the value of the square root of $\dfrac{1}{4}$.
This gives us, $\sqrt{\dfrac{1}{4}}$.
So, we are trying to simplify this term now into a simplified radical form.
Then, as, $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$ , we can write it as,
$\Rightarrow \sqrt{\dfrac{1}{4}}=\dfrac{\sqrt{1}}{\sqrt{4}}$
Again, we also know, $1={{1}^{2}}$ and $4={{2}^{2}}$ .
Putting this values, we are getting, $\sqrt{\dfrac{1}{4}}=\dfrac{\sqrt{{{1}^{2}}}}{\sqrt{{{2}^{2}}}}$
The square root is now cancelling out the square term. Then, we are getting now,
$\Rightarrow \sqrt{\dfrac{1}{4}}=\dfrac{1}{2}$.
So, we get, $\sqrt{\dfrac{1}{4}}=\pm \dfrac{1}{2}$, in a simplified radical form.
Hence, our solution is, (c) $\pm \dfrac{1}{2}$.

Note: We must take into account while solving this problem, that, $\sqrt{{{a}^{2}}}=\pm a$ . So, we will get values in the form of $\dfrac{\pm 1}{\pm 2}$ .Using the permutations of we can have,$\dfrac{1}{2},\dfrac{-1}{2},\dfrac{1}{-2},\dfrac{-1}{-2}$ . Now, after the simplification, we get, $\pm \dfrac{1}{2}$. That is how we need to get our solutions.