Question

How do you rationalize the denominator and simplify $ \dfrac{4}{\sqrt{2}} $ ?

Answer 1

Hint: In this question, we need to rationalize the denominator of $ \dfrac{4}{\sqrt{2}} $ and then simplify it. For this, we need to convert the irrational denominator $ \sqrt{2} $ into a rational number. We will multiply and divide this fraction by a certain number which can make the denominator a rational number. For denominators of the form $ \sqrt{a} $ we multiply and divide the fraction by $ \sqrt{a} $ only. After this, we will apply the property that $ \sqrt{a}\times \sqrt{a}=a $ to simplify denominator. At last we will simplify the whole fraction and get the required result.

Complete step by step answer:
Here we are given a fraction as $ \dfrac{4}{\sqrt{2}} $ . We need to rationalize the denominator of this fraction. For this we need to change the denominator of the fraction from irrational number i.e. $ \sqrt{2} $ to some rational number. We need to multiply and divide the fraction by a certain element such that the denominator becomes a rational number. For denominators of the form $ \sqrt{a} $ we multiply and divide the fraction by $ \sqrt{a} $ itself. So here we will multiply and divide the fraction by $ \sqrt{2} $ . So we get $ \dfrac{4}{\sqrt{2}}\times \dfrac{\sqrt{2}}{\sqrt{2}} $ .
In the multiplication of fractions, the numerator is multiplied with the numerator and the denominator is multiplied with the denominator.
So we have $ \dfrac{4\times \sqrt{2}}{\sqrt{2}\times \sqrt{2}} $ .
Now let us simplify the denominator. As we know that $ \sqrt{a}\times \sqrt{a} $ can be written as equal to a. So, we can write $ \sqrt{2}\times \sqrt{2} $ as 2 only. Therefore, the fraction becomes $ \dfrac{4\times \sqrt{2}}{2} $ .
Now we have rationalize the denominator. But we need to simplify the fraction also. For this, let us divide the numerator and the denominator by 2 we get $ \dfrac{4\times \sqrt{2}}{2}\div \dfrac{2}{2}\Rightarrow \dfrac{2\sqrt{2}}{1}\Rightarrow 2\sqrt{2} $ .
So our fraction $ \dfrac{4}{\sqrt{2}} $ reduces to $ 2\sqrt{2} $ which is our required answer.

Note:
Students should carefully rationalize the denominator by suitable multiplication and division. When the denominator is of the form $ a+\sqrt{b} $ then we multiply and divide the fraction by $ a-\sqrt{b} $ . We can simplify the fraction in the following way,
We have $ \dfrac{4}{\sqrt{2}} $ . We know 4 can be written as $ 2\times 2 $ . So we get $ \dfrac{2\times 2}{\sqrt{2}} $ . Further, 2 can be factorized as $ \sqrt{2}\times \sqrt{2} $ . So converting one of the 2 in the numerator we get $ \dfrac{2\times \sqrt{2}\times \sqrt{2}}{\sqrt{2}} $ . Cancelling $ \sqrt{2} $ from the numerator and denominator we get $ 2\sqrt{2} $ which is the final answer.