Simplify the following radical. [tex]$\sqrt[3]{2^3}$[/tex]
Answer (1)
We are given the expression 3 2 3 .
We use the property that n a n = a when n is odd.
Applying this property, we get 3 2 3 = 2 .
The simplified form of the radical is 2 .
Explanation
Understanding the Problem We are asked to simplify the cube root of 2 3 . This means we need to find a number that, when multiplied by itself three times, equals 2 3 , which is 2 × 2 × 2 = 8 .
Identifying the Expression The expression we need to simplify is 3 2 3 .
Applying the Property of Radicals To simplify this radical, we can use the property that n a n = a when n is odd. In our case, n = 3 and a = 2 .
Simplifying the Radical Applying this property, we have 3 2 3 = 2 .
Final Answer Therefore, the simplified form of the given radical is 2.
Examples
Imagine you are building a cube-shaped box, and you know that the volume of the box is 2 3 = 8 cubic units. To find the length of one side of the box, you need to calculate the cube root of the volume, which is 3 2 3 = 2 units. This concept is useful in various real-life situations, such as determining the dimensions of objects, calculating growth rates, and understanding scaling factors.
