Question

What is the radical form for \[{{4}^{\dfrac{1}{3}}}\]?

Answer 1

Hint: We are given with the power of cube root. In order to find the radical form of \[{{4}^{\dfrac{1}{3}}}\], we have to find such a number whose cube would result in the original number. We cannot express in the form of square root as given is of cube root. Anything written to the \[\dfrac{1}{3}\] power is the cube of the base number.

Complete step-by-step answer:
Let us learn about the radical form. Expressing a radical in its simplified form means nothing but expressing the radical in such a form that it cannot be further simplified as square root, cube root or fourth root etc. it also means that removing a radical from the fraction if exists. We simplify the radicals because simplifying the radicals by applying the same simplification rules gives us the same answer which would be easy for tallying.
Now let us find the radical form of \[{{4}^{\dfrac{1}{3}}}\].
We can find this out by quick differentiation.
\[\begin{align}
  & \sqrt{64}=\text{8 or -8} \\
 & \sqrt[3]{64}=4 \\
\end{align}\]
Since we needed the cube root, we had to find a number such that when multiplied thrice, we get the original number.
And such a number is \[4\].
\[\therefore \] The radical form of \[{{4}^{\dfrac{1}{3}}}\] is \[\sqrt[3]{4}\].

Note: The above applied method can be applied to all the powers as fractions that get smaller and smaller. Radicals can be rational numbers but all rational numbers are not radicals. Radicals that have the same root and same radicand are called radicals. Radicals can be used in everyday life in calculating areas, surface areas, total surface areas etc.