Let's convert the entire radical [tex]\(\sqrt[3]{3024}\)[/tex] into a simplified mixed radical.
1. Identify the Radicand and Factorization:
- The radicand is 3024.
2. Determine the Largest Perfect Cube that Divides the Radicand:
- We need to find the largest perfect cube that divides 3024. This value will be our [tex]\(A^3\)[/tex] where [tex]\(A\)[/tex] is the largest integer such that [tex]\(A^3\)[/tex] divides 3024 perfectly.
3. Calculate the Values:
- After determining that the largest perfect cube that divides 3024 is 216 (which is [tex]\(6^3\)[/tex]):
[tex]\[
A = 6
\][/tex]
- Now, we divide 3024 by [tex]\(6^3 = 216\)[/tex]:
[tex]\[
B = \frac{3024}{216} = 14
\][/tex]
4. Express the Mixed Radical:
- Thus, the entire radical [tex]\(\sqrt[3]{3024}\)[/tex] can be expressed as:
[tex]\[
\sqrt[3]{3024} = 6\sqrt[3]{14}
\][/tex]
The values are:
- The value of [tex]\(A\)[/tex] is [tex]\(6\)[/tex].
- The value of [tex]\(B\)[/tex] is [tex]\(14\)[/tex].
Therefore, the simplified mixed radical is:
[tex]\[
\sqrt[3]{3024} = 6 \sqrt[3]{14}
\][/tex]
