Sure! Let's determine which of the given expressions is a like radical to [tex]\(\sqrt[3]{6x^2}\)[/tex].
A like radical has the same radicand (the expression inside the radical) and the same index (the type of root) as the given radical. The given radical expression is a cube root (index of 3) with the radicand [tex]\(6x^2\)[/tex].
Let's examine each option:
1. [tex]\( x(\sqrt[3]{6x}) \)[/tex]
- This expression has a cube root (index of 3), but the radicand is [tex]\(6x\)[/tex], which is not the same as [tex]\(6x^2\)[/tex].
2. [tex]\( 6\left(\sqrt[3]{x^2}\right) \)[/tex]
- This expression has a cube root (index of 3), but the radicand is [tex]\(x^2\)[/tex], which is not the same as [tex]\(6x^2\)[/tex].
3. [tex]\( 4\left(\sqrt[3]{6x^2}\right) \)[/tex]
- This expression has a cube root (index of 3), and the radicand is [tex]\(6x^2\)[/tex], which is exactly the same as [tex]\(6x^2\)[/tex]. Therefore, this is a like radical.
4. [tex]\( x(\sqrt[3]{6}) \)[/tex]
- This expression has a cube root (index of 3), but the radicand is [tex]\(6\)[/tex], which is not the same as [tex]\(6x^2\)[/tex].
Out of these options, the correct one that is a like radical to [tex]\(\sqrt[3]{6x^2}\)[/tex] is:
[tex]\[ 4\left(\sqrt[3]{6x^2}\right) \][/tex]
Thus, the correct answer is:
[tex]\[ 4\left(\sqrt[3]{6x^2}\right) \][/tex]
