Question

How do you simplify \[\dfrac{3}{{\sqrt 3 }}\] ?

Answer 1

Hint:Simplifying the denominators with roots are often done by the method called rationalizing. To do that, we need to multiply and divide by the same root to remove the root from the denominator. In this question, we have \[\sqrt 3 \] in denominator, thus we need to multiply and divide the fraction by \[\sqrt 3 \] such that squaring of \[\sqrt 3 \] would cancel out the under root and hence the fraction would get simplified.

Complete step by step answer:
Given with a rational expression\[\dfrac{3}{{\sqrt 3 }}\] and is needed to be simplified.Generally, when we have an irrational number in a denominator, we must multiply and divide by that same irrational number. Eventually, it would not change the initial number or fraction. Because, multiply and divide by the same number would be 1 and it would not change the resultant value. And in this way, we move root from denominator to top.Multiplying and dividing by \[\sqrt 3 \], we get
\[\dfrac{3}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }} \\
\Rightarrow \dfrac{{3\sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }} \\
\Rightarrow \dfrac{{3\sqrt 3 }}{{{{(\sqrt 3 )}^2}}} \\
\Rightarrow \dfrac{{3\sqrt 3 }}{3} \\
\therefore \sqrt 3 \\ \]
Thus, the fraction gets simplified and the result is \[\sqrt 3 \].

Additional information:
If in case, in denominator there were some other factors also along with\[\sqrt 3 \], then the whole denominator is needed to be multiplied and divided by conjugate of denominator.And conjugate of denominator can be obtained by changing signs in between factors.For Example, if \[\dfrac{3}{{\sqrt 3 + 5}}\] need to be simplified, then by the method of rationalizing,
\[ \Rightarrow \dfrac{3}{{\sqrt 3 + 5}} \times \dfrac{{\sqrt 3 - 5}}{{\sqrt 3 - 5}}\]
Here, \[\sqrt 3 - 5\] is conjugate of the denominator \[\sqrt 3 + 5\].

Note:Simplify the expression as much as you can after rationalizing, by cancelling out multiples. Alternate method of solving this rational expression can be:
We know that, 3 can be written as,
\[3 = \sqrt 3 \times \sqrt 3 \]
\[ \Rightarrow \dfrac{3}{{\sqrt 3 }} = \dfrac{{\sqrt 3 \times \sqrt 3 }}{{\sqrt 3 }}\]
\[ \Rightarrow \sqrt 3 \]
On simplifying, we get resultant value =\[\sqrt 3 \].