Question

How do you simplify $\sqrt {\dfrac{5}{4}} $?

Answer 1

Hint:
Given a radical expression. We have to simplify the expression. First, we will write the expression as a quotient of two radical numbers. Then, we will find the prime factors of the numbers in the numerator and denominator. Then, make groups of prime factors in the radical expression. Then, move out a single value from the radical expression from each group. Then, write the result in simplified form.

Complete step by step solution:
We are given the expression $\sqrt {\dfrac{5}{4}} $. First, we will write the expression as a quotient of two radical terms.
\[ \Rightarrow \sqrt {\dfrac{5}{4}} = \dfrac{{\sqrt 5 }}{{\sqrt 4 }}\]
Now, we will find the prime factors of the numerator and denominator.
\[ \Rightarrow \dfrac{{\sqrt 5 }}{{\sqrt 4 }} = \dfrac{{\sqrt {5 \times 1} }}{{\sqrt {2 \times 2} }}\]
Then, move the value of the perfect square out from the radicand.
\[ \Rightarrow \dfrac{{\sqrt 5 }}{{\sqrt 4 }} = \dfrac{{\sqrt 5 }}{2}\]

Final answer: Hence, the simplified form of $\sqrt {\dfrac{5}{4}} $ is \[\dfrac{{\sqrt 5 }}{2}\]

Additional information:
The radical expression can be written as the product of two radical expression or as a quotient of two radical terms, such as \[\sqrt {a \cdot b} = \sqrt a \cdot \sqrt b \] and \[\sqrt {\dfrac{a}{b}} = \dfrac{{\sqrt a }}{{\sqrt b }}\]. Then, the square root of the denominator must be removed by applying the rationalization method. In this method, the numerator and denominator is multiplied and divided by the denominator of the given fraction. Then, the expression is simplified and the rule, \[\sqrt a \times \sqrt a = a\] is applied.

Note:
The students must note that the expression can be simplified in another way. In this method we can multiply and divide the given expression by $\sqrt 4 $such that the radical symbol can be removed from the denominator.

$ \Rightarrow \dfrac{{\sqrt 5 }}{{\sqrt 4 }} \times \dfrac{{\sqrt 4 }}{{\sqrt 4 }} = \dfrac{{\sqrt 5 \cdot \sqrt 4 }}{{\sqrt 4 \cdot \sqrt 4 }}$

Therefore, $ \Rightarrow \dfrac{{\sqrt 5 \cdot \sqrt 4 }}{4} = \dfrac{{\sqrt {20} }}{4}$

The prime factors of the numerator are
$ \Rightarrow \dfrac{{\sqrt {2 \times 2 \times 5} }}{4}$
On moving out the perfect square, we get:

$ \Rightarrow \dfrac{{2\sqrt 5 }}{4} = \dfrac{{\sqrt 5 }}{2}$
Hence, \[\dfrac{{\sqrt 5 }}{2}\] is the required answer.