Question

How do you simplify \[\dfrac{18}{\sqrt{3}}\]?

Answer 1

Hint: An irrational number is also a real number which cannot be expressed as a simple fraction or \[\dfrac{p}{q}\] form. \[\sqrt{3}\] is an irrational number. We cannot divide directly with numerator. We convert \[\sqrt{3}\] to a rational number by multiplying both the numerator and denominator with \[\sqrt{3}\] since it is under a square root we can change the denominator to rational.

Complete step by step answer:
As per the given question, we have to simplify the given expression which is the division of two rational numbers. And, the given expression is \[\dfrac{18}{\sqrt{3}}\].
Now we multiply both denominator and numerator with \[\sqrt{3}\]. Then it becomes
\[\Rightarrow \dfrac{18}{\sqrt{3}}\times \dfrac{\sqrt{3}}{\sqrt{3}}\]
The product of two irrational numbers is not always irrational but sometimes it is rational. Since in the given expression when we multiply the denominator with\[\sqrt{3}\]. Then the value under the square root becomes its square. So, we can cancel square and square roots.
\[\Rightarrow \dfrac{18}{\sqrt{{{3}^{2}}}}\times \sqrt{3}=\dfrac{18}{3}\times \sqrt{3}\]

Since we know that 18 is a multiple of 3. We can write 18 as \[3\times 6\]. Now substituting it in the expression it becomes
\[\Rightarrow \dfrac{18}{3}\times \sqrt{3}=\dfrac{6\times 3}{3}\times \sqrt{3}\]
Now we can cancel 3 in both numerator and denominator. Then the expression will be \[6\sqrt{3}\].

Therefore, the simplified form of \[\dfrac{18}{\sqrt{3}}\] is \[6\sqrt{3}\].

Note: While solving these types of problems, we need to have enough knowledge over how to convert irrational to rational numbers and how to take a conjugate of irrational numbers etc. sometimes we have complex irrational numbers then we take the conjugate of the irrational number and then simplify it. We should avoid calculation mistakes to get the correct answer.