Question

How do you write the exponential expression \[3{{x}^{\dfrac{3}{8}}}\] in radical form?

Answer 1

Hint: First understand the meaning of the term ‘radical form of a number’. Now, to write the radical form of the given exponential expression, first write the exponent \[\dfrac{3}{8}\] as \[3\times \dfrac{1}{8}\]. In the next step use the conversion \[{{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}\] to get the answer.

Complete step by step answer:
Here, we have been provided with the exponential expression \[3{{x}^{\dfrac{3}{8}}}\] and we are asked to write it in the radical form. But first we need to understand the term ‘radical’.
In mathematics, a radical is also called the root. Roots are defined as the special case of exponentiation where the exponent is a fraction. Radical or root is generally denoted by the sign \[\sqrt[n]{{}}\]. For example: - let us consider a number x then its \[{{n}^{th}}\] root is given as \[\sqrt[n]{x}\] in radical form. Here, x is called the radicand. If n is positive then x must be positive for real roots to occur.
Now, in exponential form we write the exponent as a fraction. For example: - the radical expression \[\sqrt[n]{x}\] is written as \[{{x}^{\dfrac{1}{n}}}\] in exponential form. Here, \[\dfrac{1}{n}\] represents the inverse of n.
Let us come to the question. We have the expression \[3{{x}^{\dfrac{3}{8}}}\]. Now, we can write this expression as: -
\[\Rightarrow 3{{x}^{\dfrac{3}{8}}}=3\times {{x}^{3\times \dfrac{1}{8}}}\]
Here, considering n = 8, we can write the radical form as: -
\[\Rightarrow 3{{x}^{\dfrac{3}{8}}}=3\times \sqrt[8]{{{x}^{3}}}\]
Here, \[\sqrt[8]{{{x}^{3}}}\] denotes the \[{{8}^{th}}\] root of \[{{x}^{3}}\].
Hence, \[3\times \sqrt[8]{{{x}^{3}}}\] is the radical form of the given exponential expression and our answer.

Note:
One may note that here the exponent 3 will remain as it is because as a numerator in the exponent \[\dfrac{3}{8}\]. So, in general if the exponent is of the form \[\dfrac{a}{b}\] then we have to write it as \[a\times \dfrac{1}{b}\] and then in the radical form it is written as \[\sqrt[b]{{{x}^{a}}}\], i.e., \[{{b}^{th}}\] root of \[{{x}^{a}}\]. You must remember the difference between the representations of exponential form and radical form of a number. Remember that \[\sqrt{{}}\] denotes square root, \[\sqrt[3]{{}}\] denotes cube root, \[\sqrt[4]{{}}\] denotes \[{{4}^{th}}\] root and so on.