Question

How do you write ${x^{\dfrac{2}{3}}}$ in radical form ?

Answer 1

Hint:
Radicals means roots. Symbol of root $\sqrt {} $ is known as a radical symbol. If $n$ is a positive integer that is greater than $m$ and $a$ is real number then , ${a^{\dfrac{m}{n}}}$ can be written as $\sqrt[n]{{{a^m}}}$
Here $n$ is denoted as the index, $a$ is denoted as the radicand and we already know the symbol $\sqrt {} $ is called the radical.
The left side of the equation below is called the radical form and the right side is called the exponent form as we have removed the symbol $\sqrt {} $ .
In this question we have to convert the exponent form in radical form.

Complete step by step solution:
Firstly we will write ${x^{\dfrac{2}{3}}}$ in simplest form to make it easier to solve.
Simplify the fraction like this ${a^{\dfrac{m}{n}}} = {({a^{\dfrac{1}{n}}})^m}$
You can break the fraction $\dfrac{2}{3}$ in this form $2 \times \dfrac{1}{3}$
Now, putting the above in the form of power like this,
${x^{2 \times \dfrac{1}{3}}}$
Here you can say that fraction $\dfrac{1}{3}$ means cube root of x , writing ${x^{\dfrac{2}{3}}}$ in radical form
Take $a$ as $x$ , $m$ as 2 and $n$ as 3
Putting the above values in $\sqrt[n]{{{a^m}}}$
$ \Rightarrow \sqrt[3]{{{{(x)}^2}}}$

Thus, $\sqrt[3]{{{{(x)}^2}}}$ is a radical form of ${x^{\dfrac{2}{3}}}$.

Note:
All exponents in radicand should be less than index, for example in this question radicand is $x$ , 2 is exponent of radicand $x$ and 3 is index. If index is greater than exponent of then the simplified form of that will not be a fraction which cannot be written as radical form.
In radical form there is no fraction in the power of radicand , if there is a fraction in power of radicand that means you have not simplified it properly. Try to solve using brackets to avoid confusion.