Question

How do you write \[{u^{\dfrac{3}{8}}}\] as a radical expression?

Answer 1

Hint: Here, we will write the fractional exponent with the radical sign by using the rules of the fractional exponent. The radical form can be written only with the Radical sign. The radical symbol or sign represents the square root of a number or a variable.

Formula Used:
Rule of Radicals: \[\sqrt[k]{n} = {n^{\dfrac{1}{k}}}\]

Complete step by step solution:
We are given a variable \[{u^{\dfrac{3}{8}}}\].
We know that the given variable is in the form of the exponential expression.
Now, we will rewrite the given variable as a Radical expression.
We know that usually the Radical of a number \[n\] is denoted by the symbol \[\sqrt n \].
We know that the Radical symbol represents the \[{k^{th}}\] root in the form of fractional exponents.
Rule of Radicals: \[\sqrt[k]{n} = {n^{\dfrac{1}{k}}}\]
Now, by using the rule of Radicals, we get
\[{\left( u \right)^{\dfrac{3}{8}}} = {\left( {\sqrt[8]{u}} \right)^3}\]
Now, we will rewrite the variable, so we get
\[ \Rightarrow {\left( u \right)^{\dfrac{3}{8}}} = \left( {\sqrt[8]{{{u^3}}}} \right)\]

Therefore, the Radical form of \[{u^{\dfrac{3}{8}}}\] is \[\sqrt[8]{{{u^3}}}\].

Additional information:
If the number or a variable has a fractional exponent, then the numerator of a fractional exponent represents the power of the radical form and the denominator of a fractional exponent represents the index of the radical form. We can write power at first and then the index or we can write index at first and then the power. This can be written in either way to express the number in the Radical form.

Note:
We should note that the radical sign indicates to find the root of a number or a variable. We know that the radical sign with a small number \[n\] is known as \[n\] th root of a number or a variable. The smaller number is called the index. We should also remember that usually the square root of a number or a variable is not written with any index. In simple, we write the fractional exponent of a number or a variable \[{x^{\dfrac{p}{q}}}\] in the Radical form as \[\sqrt[q]{{{x^p}}}\] or \[{\left( {\sqrt[q]{x}} \right)^p}\]. The radical form can be written only with the Radical sign.