To find which expression among the given options is a like radical to [tex]\(\sqrt[3]{6 x^2}\)[/tex], we need to understand what each expression means and see which one simplifies to [tex]\(\sqrt[3]{6 x^2}\)[/tex].
Let's start with the given expression:
[tex]\[ \sqrt[3]{6x^2} \][/tex]
Now, let's examine each option one by one:
### Option 1: [tex]\( x(\sqrt[3]{6 x}) \)[/tex]
First, simplify the expression inside the parentheses:
[tex]\[ \sqrt[3]{6 x} = (6x)^{1/3} \][/tex]
Then, multiply by [tex]\( x \)[/tex]:
[tex]\[ x \cdot (6x)^{1/3} \][/tex]
[tex]\[ x \cdot 6^{1/3} x^{1/3} \][/tex]
[tex]\[ 6^{1/3} x^{1 + 1/3} \][/tex]
[tex]\[ 6^{1/3} x^{4/3} \][/tex]
This simplifies to:
[tex]\[ 6^{1/3} x^{4/3} \][/tex]
This is not the same as the original radical [tex]\(\sqrt[3]{6x^2}\)[/tex].
### Option 2: [tex]\( 6(\sqrt[3]{x^2}) \)[/tex]
Simplify the expression inside the parentheses:
[tex]\[ \sqrt[3]{x^2} = x^{2/3} \][/tex]
Then, multiply by 6:
[tex]\[ 6 \cdot x^{2/3} \][/tex]
[tex]\[ 6 x^{2/3} \][/tex]
This is not the same as the original radical [tex]\(\sqrt[3]{6x^2}\)[/tex].
### Option 3: [tex]\( 4(\sqrt[3]{6 x^2}) \)[/tex]
This expression is simply:
[tex]\[ 4(\sqrt[3]{6x^2}) \][/tex]
Since we want to see if the expression is a like radical to [tex]\(\sqrt[3]{6x^2}\)[/tex], we recognize that multiplying [tex]\(\sqrt[3]{6x^2}\)[/tex] by 4 changes the coefficient but not the radical form itself. The radical part remains [tex]\(\sqrt[3]{6x^2}\)[/tex], so this is not a like radical; it's a scaled version.
### Option 4: [tex]\( x(\sqrt[3]{6}) \)[/tex]
Simplify the expression inside the parentheses:
[tex]\[ \sqrt[3]{6} = 6^{1/3} \][/tex]
Then, multiply by [tex]\( x \)[/tex]:
[tex]\[ x \cdot 6^{1/3} \][/tex]
[tex]\[ 6^{1/3} x \][/tex]
This is not the same as the original radical [tex]\(\sqrt[3]{6x^2}\)[/tex].
After analyzing all the options, none of them exactly matches the form of the original radical [tex]\(\sqrt[3]{6x^2}\)[/tex].
Therefore, the correct answer is that none of the options are a like radical to [tex]\(\sqrt[3]{6x^2}\)[/tex]. However, within the context of the given multiple-choice question, the closest is Option 2: [tex]\( 6(\sqrt[3]{x^2}) \)[/tex] because it keeps some of the original form, but this does not strictly satisfy the condition of being an exact like radical.
