Question

What is the square root of $\dfrac{3}{4}$ $?$

Answer 1

Hint: To solve this question we need to know the concept of exponents and powers. In this question we will prime factorise the number to find the $\dfrac{1}{2}th$ power of the number. We will use the Prime Factorization method to get the factors of the number, this helps in finding the factors of the number easily. We can recheck the solution by solving the answer having its power as $2$ or squaring it which we get after solving.

Complete step-by-step solution:
The question asks us to simplify or evaluate ${{\left( \dfrac{3}{4} \right)}^{\dfrac{1}{2}}}$ . This is a question of exponent. We need to convert the number $\dfrac{3}{4}$ into all the prime factors. To find the $\dfrac{1}{2}th$ of the number we write the number $4$ as the product of all the prime factors associated with it as we know that $3$is a prime factor so it will not have any factor other than $1$ and $3$ itself.
We can find the value of factors by prime factorisation of the given number. So it would be written as,
\[4=2\times 2\]
Now, on substituting $\dfrac{3}{4}$ with the product of its prime factors, we get:
\[\Rightarrow {{\left( \dfrac{3}{4} \right)}^{\dfrac{1}{2}}}\]
The above function could be written as
\[\Rightarrow {{\left( \dfrac{3}{2\times 2} \right)}^{\dfrac{1}{2}}}\]
We can evaluate the number by finding the square root.
For finding the square root if we have a number $b=a\times a$, then it is the square root of $b$, $\sqrt{b}=\sqrt{a\times a}$ , which is equal to $a$ . So applying the same, on the given value:
$\Rightarrow {{\left( \dfrac{3}{2\times 2} \right)}^{\dfrac{1}{2}}}$
$\Rightarrow \dfrac{\sqrt{3}}{2}$
Value of $\sqrt{3}$ in decimal form is $1.732$, applying the same in the formula we get:
$\Rightarrow \dfrac{1.732}{2}$
On further calculating the above expression we get:
$\Rightarrow 0.866$
$\therefore $ The square root of $\dfrac{3}{4}$ is $\dfrac{\sqrt{3}}{2}$ which is $0.866$.

Note: We can check whether the answer is correct or not. To check this we will find ${{\left( \dfrac{\sqrt{3}}{2} \right)}^{2}}$ and check whether the answer matches the question or not. Let us square the value $\dfrac{\sqrt{3}}{2}$, the result we get after squaring is
\[\Rightarrow \dfrac{\sqrt{3}\times \sqrt{3}}{2\times 2}\]
On multiplying the numerator and the denominators together we get,
\[\Rightarrow \dfrac{3}{4}\]
So the value which we get is the same as the question. So our solving is correct for the question.