Model the mixed radical $2 \sqrt[3]{4}$ as an entire radical. Record the value of the radicand.
Answer (1)
Rewrite the coefficient 2 as a cube root: 2 = 3 2 3 .
Combine the radicals using the property n a ⋅ n b = n a ⋅ b : 3 2 3 ⋅ 3 4 = 3 2 3 ⋅ 4 .
Simplify the expression: 3 8 ⋅ 4 = 3 32 .
The radicand of the entire radical is 32 .
Explanation
Understanding the Problem We are given the mixed radical 2 3 4 and we want to express it as an entire radical, which means we want to write it in the form 3 x for some value x . This value x is called the radicand.
Moving the Coefficient Inside the Radical To convert the mixed radical to an entire radical, we need to move the coefficient 2 inside the cube root. Since we are dealing with a cube root, we need to cube the coefficient before placing it inside the radical. So, we rewrite 2 as 3 2 3 .
Combining the Radicals Now we have 2 3 4 = 3 2 3 ⋅ 3 4 . Using the property of radicals that n a ⋅ n b = n a ⋅ b , we can combine the two cube roots: 3 2 3 ⋅ 3 4 = 3 2 3 ⋅ 4 = 3 8 ⋅ 4 = 3 32 .
Finding the Radicand Therefore, the mixed radical 2 3 4 can be written as the entire radical 3 32 . The radicand is the value inside the cube root, which is 32.
Examples
Understanding radicals is crucial in fields like engineering and physics, where you often deal with complex calculations involving roots. For instance, when calculating the volume of a cube with side length 2 3 4 , expressing it as an entire radical simplifies the computation. By converting it to 3 32 , you can easily cube the side length to find the volume: ( 3 32 ) 3 = 32 . This skill is also useful in simplifying expressions in thermodynamics or fluid dynamics, where radical expressions frequently appear.
