To determine which of the given radicals is like the radical [tex]\(\sqrt[3]{6x^2}\)[/tex], we need to identify the radical that has the same radicand (the expression inside the radical) and the same index (the root).
1. [tex]\(x(\sqrt[3]{6x})\)[/tex]:
- Radicand: [tex]\(6x\)[/tex]
- Index: 3 (cube root)
- This radicand [tex]\(6x\)[/tex] is different from [tex]\(6x^2\)[/tex]. Therefore, this is not a like radical.
2. [tex]\(6\left(\sqrt[3]{x^2}\right)\)[/tex]:
- Radicand: [tex]\(x^2\)[/tex]
- Index: 3 (cube root)
- This radicand [tex]\(x^2\)[/tex] is different from [tex]\(6x^2\)[/tex]. Therefore, this is not a like radical.
3. [tex]\(4\left(\sqrt[3]{6x^2}\right)\)[/tex]:
- Radicand: [tex]\(6x^2\)[/tex]
- Index: 3 (cube root)
- This has the same radicand [tex]\(6x^2\)[/tex] and the same index 3, so it is a like radical.
4. [tex]\(x(\sqrt[3]{6})\)[/tex]:
- Radicand: [tex]\(6\)[/tex]
- Index: 3 (cube root)
- This radicand [tex]\(6\)[/tex] is different from [tex]\(6x^2\)[/tex]. Therefore, this is not a like radical.
From the given options, the only radical that matches both the radicand [tex]\(6x^2\)[/tex] and the index 3 is:
[tex]\[4\left(\sqrt[3]{6x^2}\right)\][/tex]
Therefore, the like radical to [tex]\(\sqrt[3]{6x^2}\)[/tex] is [tex]\(4\left(\sqrt[3]{6x^2}\right)\)[/tex] and the correct option is:
3
