Question

How do you simplify $\dfrac{3}{{\sqrt 2 }}$?

Answer 1

Hint:
Here we basically need to convert the irrational number which is in the denominator into the rational number by rationalizing the denominator. We need to multiply the irrational number present in the denominator in the numerator and the denominator again and we will get the rational number in the denominator and we will be able to simplify the given expression.

Complete step by step solution:
Here we are given that we need to simplify the expression which is given as $\dfrac{3}{{\sqrt 2 }}$
In such types of problems of simplification where we are given the irrational number at the denominator, we need to convert it into rational by rationalizing the denominator.
For this, we must know what the difference between rational and irrational numbers is. So basically rational numbers are those which can be written in the form $\dfrac{p}{q}{\text{ and }}p,q \in Z$
For example: $\dfrac{2}{7},\dfrac{2}{1} = 2$ all are rational numbers while irrational are those that cannot be represented in such manner. For example:$\sqrt 2 ,\sqrt 3 $
Now we need to just rationalize the denominator in the given term in order to get the simplified form of that number.
We are given:
$\dfrac{3}{{\sqrt 2 }}$
We need to multiply the irrational number in the denominator once again in the numerator and denominator and we will get:
$\dfrac{3}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}$
Now we know that multiplication of two same digits under the root gives us the digit itself which means $\sqrt a .\sqrt a = a$
So we can substitute $\sqrt 2 .\sqrt 2 = 2$ in the above denominator and we will get:
$\dfrac{3}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }} = \dfrac{{3\sqrt 2 }}{2}$
So we get the simplified form of $\dfrac{3}{{\sqrt 2 }}$ as $\dfrac{{3\sqrt 2 }}{2}$

Note:
Here the student can be asked to simplify the term of the form $\dfrac{7}{{\sqrt 3 - \sqrt 2 }}$ also. So here we need to multiply the denominator with the same denominator again but opposite in sign. So the minus sign will change into plus and vice versa.
Now we can apply the formula $(a - b)(a + b) = {a^2} - {b^2}$ in the denominator and get the desired result.