Question

Write an equivalent exponential form for radical expression.
\[\sqrt[3]{{13}}\]

Answer 1

Hint:
In the given question, we are required to convert the given radical expression into its equivalent exponential form. To do that, we will identify the base in the expression. Then, we will identify the power and raise the base to that power.

Complete step by step answer:
To solve this question, the students must know about the concept of radical expressions and exponential expressions.
Radical expressions are those expressions that contain the radical symbol \['\sqrt[n]{{}}'\]. The power is written outside the radical symbol, in place of the \[n\]. The base is written inside the radical. The radical represents the power in the form \[\dfrac{1}{n}\].
Exponential expressions are the expressions that can be expressed in the form \[{A^b}\], where, \[A\] is the base and \[b\] is the power.
In the given question, the base will be the number that is inside the radical symbol. Since the number given is 13 inside the radical, our base is 13.
Now, our power will be the number that is given outside the radical symbol \[\sqrt[n]{{}}\]. As we are given 3 outside the radical symbol here, our power will be 3. We can also conclude this to be a cube-root.
Now, we know that exponentials are represented by \[{A^b}\].
Substituting 13 for \[A\] and \[\dfrac{1}{3}\] for \[b\] in the expression \[{A^b}\], we get \[{13^{\dfrac{1}{3}}}\].
Thus, the equivalent exponential form of radical expression \[\sqrt[3]{{13}}\] is \[{13^{\dfrac{1}{3}}}\].

Note:
It is important to note that in the case of power two, the radical is known as square-root. For square-root, we do not mention power. But, for higher numbers, the power is shown out of the radical symbol. Hence the radical symbol should not be confused with square-root, because it is a type of radical only. Also, the radical symbol \[\sqrt[n]{{}}\]represents the power as \[\dfrac{1}{n}\], and not \[n\].