To determine which of the given options is a like radical to [tex]\(\sqrt[3]{7x}\)[/tex], we need to identify the radicals that have the same radicand (the expression under the radical) and the same index (the root's degree). Here's the step-by-step analysis of each option:
1. Option 1: [tex]\(4(\sqrt[3]{7x})\)[/tex]
- Radicand: [tex]\(7x\)[/tex]
- Index: [tex]\(3\)[/tex]
- Coefficient: The number "4" is a coefficient and does not affect the classification of the radical.
2. Option 2: [tex]\(\sqrt{7x}\)[/tex]
- Radicand: [tex]\(7x\)[/tex]
- Index: This is a square root, which has an index of [tex]\(2\)[/tex].
3. Option 3: [tex]\(x(\sqrt[3]{7})\)[/tex]
- Radicand: [tex]\(7\)[/tex]
- Index: [tex]\(3\)[/tex]
- Note: The variable [tex]\(x\)[/tex] is outside the cube root and thus does not affect the radicand.
4. Option 4: [tex]\(7 \sqrt{x}\)[/tex]
- Radicand: [tex]\(x\)[/tex]
- Index: This is a square root, which has an index of [tex]\(2\)[/tex].
- Coefficient: The number "7" is a coefficient and does not affect the classification of the radical.
For a radical expression to be a like radical to [tex]\(\sqrt[3]{7x}\)[/tex], both the radicand and the index must match. Let's compare each option with the given radical [tex]\(\sqrt[3]{7x}\)[/tex]:
- Option 1 matches because it has the same radicand ([tex]\(7x\)[/tex]) and the same index ([tex]\(3\)[/tex]).
- Option 2 does not match because, even though the radicand ([tex]\(7x\)[/tex]) is the same, the index ([tex]\(2\)[/tex]) is different.
- Option 3 does not match because, although the index ([tex]\(3\)[/tex]) is the same, the radicand ([tex]\(7\)[/tex]) is different.
- Option 4 does not match because both the radicand ([tex]\(x\)[/tex]) and the index ([tex]\(2\)[/tex]) are different.
Thus, the only like radical to [tex]\(\sqrt[3]{7x}\)[/tex] among the given options is:
[tex]\[ 4(\sqrt[3]{7x}) \][/tex]
Therefore, the correct option is option 1.
