Question

What is the square root of \[24\] / square root of \[6\] ?

Answer 1

Hint: According to the question; we have to find the value of the square root of \[24\] / square root of \[6\] i.e., \[\dfrac{{\sqrt {24} }}{{\sqrt 6 }}\] . In order to solve this question, first we will use the property given by \[\dfrac{{\sqrt x }}{{\sqrt y }} = \sqrt {\dfrac{x}{y}} \] and write the given term as \[\sqrt {\dfrac{{24}}{6}} \] . Then we will rewrite the numbers as the product of their prime factors. After that we will cancel the common factors from the numerator and denominator and find the square root of the obtained value.Hence, we will get the required value of the given expression.

Complete step by step answer:
According to the question we have to find the value of is the square root of \[24\] / square root of \[6\] i.e., we have to simplify the value of \[\dfrac{{\sqrt {24} }}{{\sqrt 6 }}\]. Now we know that
\[\dfrac{{\sqrt x }}{{\sqrt y }} = \sqrt {\dfrac{x}{y}} \]
According to the question,
\[x = 24\] and \[y = 6\]
Therefore, we can write
\[\dfrac{{\sqrt {24} }}{{\sqrt 6 }} = \sqrt {\dfrac{{24}}{6}} {\text{ }} - - - \left( i \right)\]

Now we will rewrite \[24\] and \[6\] as the product of their prime factors.
Therefore, we have
\[24 = 2 \times 2 \times 2 \times 3\]
And \[6 = 2 \times 3\]
Now on substituting the values in the equation \[\left( i \right)\] we have
\[\dfrac{{\sqrt {24} }}{{\sqrt 6 }} = \sqrt {\dfrac{{24}}{6}} = \sqrt {\dfrac{{2 \times 2 \times 2 \times 3}}{{2 \times 3}}} \]
Here we can see the numerator and denominator have the common factor as \[2 \times 3\].

So, on cancelling the common factor, we get
\[ \Rightarrow \dfrac{{\sqrt {24} }}{{\sqrt 6 }} = \sqrt {\dfrac{{2 \times 2}}{1}} \]
On solving, we get
\[ \Rightarrow \dfrac{{\sqrt {24} }}{{\sqrt 6 }} = \sqrt 4 \]
Now we know, \[\sqrt 4 = \pm 2\]
Therefore, we have
\[ \therefore \dfrac{{\sqrt {24} }}{{\sqrt 6 }} = \pm 2\]

Hence, the value of \[\dfrac{{\sqrt {24} }}{{\sqrt 6 }}\] is \[ \pm 2\].

Note: We have used the term “prime factors” in the solution. So, a prime factor is a factor that is a prime number. Also note that to simplify the terms inside the radicals, try to factor them to find at least one term that is a perfect square. Also keep in mind when a square root is found we must add the plus-or-minus sign because we have \[{\left( 2 \right)^2} = 4\] and \[{\left( { - 2} \right)^2} = 4\].