Question

How do I rationalize Cubic roots?

Answer 1

Hint: This type of question is based on the concept of rationalizing. Here, to explain the concept let us consider a cubic root \[\dfrac{a}{\sqrt[3]{b}}\]. Here, \[\sqrt[3]{b}\] is a cubic term of b. We have to multiply \[\sqrt[3]{{{b}^{2}}}\] in both the numerator and denominator to rationalise the cubic root. Using \[\sqrt[3]{{{b}^{3}}}=b\], we can simplify the cubic root \[\dfrac{a}{\sqrt[3]{b}}\]. When we get ‘b’ alone in the denominator, we have rationalised the cubic root considered. Thus, we got the final answer.

Complete step by step answer:
According to the question, we are asked to find how to rationalize a cubic root.
Let us assume \[\dfrac{a}{\sqrt[3]{b}}\] to be a cubic root.
Let us now consider \[\dfrac{a}{\sqrt[3]{b}}\].
Now, let us multiply \[\sqrt[3]{{{b}^{2}}}\] in both the numerator and denominator of \[\dfrac{a}{\sqrt[3]{b}}\].
We get,
\[\dfrac{a}{\sqrt[3]{b}}=\dfrac{a}{\sqrt[3]{b}}\times \dfrac{\sqrt[3]{{{b}^{2}}}}{\sqrt[3]{{{b}^{2}}}}\]
On grouping the numerator and denominator, we get,
\[\Rightarrow \dfrac{a}{\sqrt[3]{b}}=\dfrac{a\sqrt[3]{{{b}^{2}}}}{\sqrt[3]{b}\times \sqrt[3]{{{b}^{2}}}}\]
We know that \[\sqrt[3]{x}\times \sqrt[3]{y}=\sqrt[3]{xy}\].
On using this property of cubic root in the above obtained expression, we get,
\[\Rightarrow \dfrac{a}{\sqrt[3]{b}}=\dfrac{a\sqrt[3]{{{b}^{2}}}}{\sqrt[3]{{{b}^{3}}}}\] ---------(1)
We know that \[\sqrt[3]{{{b}^{3}}}=b\].
Using this in equation (1), we get
\[\dfrac{a}{\sqrt[3]{b}}=\dfrac{a\sqrt[3]{{{b}^{2}}}}{b}\]
Therefore, we can rationalize the cubic root by multiplying \[\sqrt[3]{{{b}^{2}}}\] on both the numerator and denominator of the assumed cubic roots.
Hence, the cubic root \[\dfrac{a}{\sqrt[3]{b}}\] has been rationalised.

Note:
We must always multiply numerator and denominator with the cube root of the square of the term in the denominator to rationalise. We can rationalize negative cubic root also by the same way. Similarly, we can rationalize \[\dfrac{2}{\sqrt[3]{7}}\]. Here a=2 and b=7. Follow the above steps to rationalise the cubic root. We should know the square of basic numbers for easy calculations. We should avoid calculation mistakes based on sign conventions, if any.