Question

How do you write the expression \[{{2}^{\dfrac{5}{3}}}\] in radical form?

Answer 1

Hint: To solve the given question we will need the following properties. The property of exponents which states that \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\] here, \[a,m,n\in \]Real numbers. We should also know that if a is the 5th power of b, then it can be written as \[a={{b}^{5}}\], we will use these properties to solve the given question.

Complete step by step answer:
We are asked to simplify \[{{2}^{\dfrac{5}{3}}}\], which means we have to find its value. We know that 32 is the 5th power of 2. Using the property which states that, if a is the 5th power of b, then it can be written as \[a={{b}^{5}}\]. Here we have a = 32, and b = 2. By substituting the value, it can be written as, \[32={{2}^{5}}\]. Using this in the given question, it can be simplified as,
\[\Rightarrow {{32}^{\dfrac{1}{3}}}\]
To get rid of the cube root, the term inside it should be a perfect cube. As we know that the 32 is not a perfect cube, we can not get rid of the cube root.
So, on simplifying we can write the given expression as \[{{32}^{\dfrac{1}{3}}}\]

Note: These types of questions can be solved by remembering the values of squares, cubes, square roots, and cube roots of the numbers.
Here, we can also find the approximate value of the cube root of 32 using a calculator. If the question is to evaluate the value. But here we are just asked to express in radical form so we don’t need to evaluate its value.