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How do you simplify $\dfrac{1}{{\sqrt 3 }}$?

seo-qna
Last updated date: 27th Feb 2024
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IVSAT 2024
Answer
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Hint: Here, we are asked to simplify the fraction whose denominator is a square root value. Thus, ultimately we have to rationalize the denominator in this question. Rationalization is defined as the process of removing the radical or imaginary number from the denominator of a fraction. Generally, the denominator of a fraction can be rationalized by taking the conjugate of a denominator and multiplying it by both the numerator and denominator.

Complete step-by-step solution:
We are given the value $\dfrac{1}{{\sqrt 3 }}$.
We will now rationalize its denominator. We will do it by removing $\sqrt 3 $from the denominator, and replace it with a whole number.
For this we will divide and multiply the given number with $\sqrt 3 $.
$ \Rightarrow \dfrac{1}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }} = \dfrac{{\sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }}$
We know that when a number is multiplied two time by itself then it will give the square of the number.
Therefore, here in the denominator, $\sqrt 3 $ is multiplied with $\sqrt 3 $which will give us its square value which is 3. This means that $\sqrt 3 \times \sqrt 3 = 3$.
Thus, we can write
$ \Rightarrow \dfrac{1}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }} = \dfrac{{\sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }} = \dfrac{{\sqrt 3 }}{3}$
Thus, by simplifying $\dfrac{1}{{\sqrt 3 }}$, we get $\dfrac{{\sqrt 3 }}{3}$.

Note: Here, we have done nothing but converted the denominator into the whole number. It does not mean that it is wrong to write $\dfrac{1}{{\sqrt 3 }}$but it is the standard form is to avoid roots in denominators, if possible. Moreover, rationalizing the denominator is a useful skill that can help us to solve problems in higher mathematics. For example, in calculus, it is a trick that can often help us to solve some problems about limits as well as complex numbers.