Question

How do you write \[{81^{\dfrac{3}{4}}}\] in Radical form?

Answer 1

Hint: Here, we will use the definition of radical of a number. We will then write the fractional exponent with the radical sign by using the rules of radicals for the fractional exponent. If a number 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.

Complete Step by Step Solution:
We are given a number \[{81^{\dfrac{3}{4}}}\].
Let \[x = {81^{\dfrac{3}{4}}}\]
We know that the given number is in the exponential form. Now, we will rewrite the given number in a Radical form.
We know that the Radical of a number \[n\] is denoted by the symbol \[\sqrt n \]. We also know that the radical symbol represents the square root of a number.
The radical symbol represents the \[{k^{th}}\] root in the form of fractional exponents.
So, we get \[\sqrt[k]{n} = {n^{\dfrac{1}{k}}}\].
Now, by rewriting the number using the rule of Radicals, we get
\[ \Rightarrow {\left( {81} \right)^{\dfrac{3}{4}}} = {\left( {\sqrt[4]{{81}}} \right)^3}\]
Now, we will rewrite the number, so we get
\[ \Rightarrow {\left( {81} \right)^{\dfrac{3}{4}}} = \left( {\sqrt[4]{{{{81}^3}}}} \right)\]

Therefore, the Radical form of \[{81^{\dfrac{3}{4}}}\]is\[\sqrt[4]{{{{81}^3}}}\].

Additional information:
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. In simple, we write the fractional exponent of a number \[{x^{\dfrac{p}{q}}}\] 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.

Note:
We should note that the radical sign indicates to find the root of a number. We know that the radical sign with a small number \[n\] is known as \[{n^{th}}\] root of a number. The smaller number is called the index. We should also remember that usually the square root of a number is not written with any index. The given fractional exponent can be simplified into an integer. Here, we need to keep in mind that we have to represent the equation in the radical form and not simplify the expression.