Question

How do you express ${x^{\dfrac{4}{3}}}$ in simplest radical form?

Answer 1

Hint: In this question, we are given an algebraic unit and we have been asked to express it in the simplest radical form. In order to answer this question, you must be aware about the rules of exponents. Split the power such that the numerator is being multiplied with the denominator. Use the exponent rules of ${\left( {{x^a}} \right)^b}$ and write the given unit in this form. Then find the ${b^{th}}$ root from here and this will be your answer.

Formula used: Exponent rule: ${\left( {{x^a}} \right)^b} = {x^{ab}}$

Complete step-by-step solution:
We are given ${x^{\dfrac{4}{3}}}$ and we have been asked to express it in the simplest radical form. Let us focus on the power.
The power of the given number is $\dfrac{4}{3}$ .
We can write this number as a product of the numerator and the denominator in the following way:
$ \Rightarrow 4 \times \dfrac{1}{3}$
Looking at the exponent rule ${\left( {{x^a}} \right)^b} = {x^{ab}}$ , we can assume $a = 4$ and $b = \dfrac{1}{3}$ .
Putting the number back in the given number,
$ \Rightarrow {x^{4 \times \dfrac{1}{3}}}$
Using the rule of the exponents, we can also write it as –
$ \Rightarrow {\left( {{x^4}} \right)^{\dfrac{1}{3}}}$
Now, if the power of the number is $\dfrac{1}{3}$ , it simply means that we have to find the $3rd$ root of the number, or we can say that we have to find the cube root of the given number.

We will write it as $ \Rightarrow \sqrt[3]{{{x^4}}}$

Note: Since we only need to arrange the power in the form of the exponent rule, we can also write it in the following way:
We can also assume $a = \dfrac{1}{3}$ and $b = 4$ .
Now, in power form, it can be written as –
$ \Rightarrow {\left( {{x^{\dfrac{1}{3}}}} \right)^4}$
Now, we will write the $3rd$ root of the $x$, and not of ${x^4}$ . See how it is done –
$ \Rightarrow {\left( {\sqrt[3]{x}} \right)^4}$
Hence, the given number can also be simplified in this way.