Question

Find the rationalising factor of the given binomial surd: ${2^{\dfrac{1}{3}}} + {2^{ - \dfrac{1}{3}}}$

Answer 1

Hint: In this question we have to find the rationalising factor of the given expression. In order to rationalize we will firstly, take LCM and then rationalize. This helps us simplify the expression and eventually reach the answer.

Complete step-by-step answer:
In this question we have been given the binomial surd: ${2^{\dfrac{1}{3}}} + {2^{ - \dfrac{1}{3}}}$
Which can also be written as, ${2^{\dfrac{1}{3}}} + \dfrac{1}{{{2^{\dfrac{1}{3}}}}}$
Now if we take LCM then we get,
$ \Rightarrow \dfrac{{{2^{\dfrac{2}{3}}} + 1}}{{{2^{\dfrac{1}{3}}}}}$
Now, in rationalization we have to get the denominator in integral form.
So, multiply and divide the expression with ${2^{\dfrac{2}{3}}}$
$ \Rightarrow \dfrac{{{2^{\dfrac{2}{3}}} + 1}}{{{2^{\dfrac{1}{3}}}}} = \dfrac{{{2^{\dfrac{2}{3}}}\left( {{2^{\dfrac{2}{3}}} + 1} \right)}}{{{2^{\dfrac{1}{3} + \dfrac{2}{3}}}}}$
$ \Rightarrow \dfrac{{{2^{\dfrac{4}{3}}} + {2^{\dfrac{2}{3}}}}}{{{2^{\dfrac{3}{3}}}}} = \dfrac{{{2^{\dfrac{4}{3}}} + {2^{\dfrac{2}{3}}}}}{2}$
As, the expression got rationalised by multiplying and dividing with ${2^{\dfrac{2}{3}}}$, so the rationalising factor is ${2^{\dfrac{2}{3}}}$.

Note: Whenever we face such types of problems the value point to remember is that we need to have a good grasp over surds and exponents. We must also firstly do the LCM in order to find the rationalising factor. This helps in getting us the required expressions and gets us on the right track to reach the answer.