Let's analyze each of the given equations to determine which one is a radical equation.
1. [tex]\( 4p = \sqrt{-3} + p \)[/tex]
- The term [tex]\(\sqrt{-3}\)[/tex] is a radical, but it does not involve a variable.
- There are no variables under the square root or any other root.
- Therefore, this is not considered a radical equation.
2. [tex]\( x \sqrt{3} + x = \sqrt[3]{2x} \)[/tex]
- [tex]\( x \sqrt{3} \)[/tex] involves a product of [tex]\( x \)[/tex] and the square root of a constant, which is not a radical term in the true sense of a radical equation.
- However, [tex]\( \sqrt[3]{2x} \)[/tex] involves a variable [tex]\( x \)[/tex] under the cube root.
- Since there is a variable under the cubic root, this equation qualifies as a radical equation.
3. [tex]\( 7 \sqrt{11} - w = -34 \)[/tex]
- The term [tex]\(\sqrt{11}\)[/tex] is a radical, but it does not involve a variable.
- There are no variables under the square root or any other root.
- Therefore, this is not considered a radical equation.
4. [tex]\( 5 - \sqrt[3]{8} = \gamma \sqrt{16} \)[/tex]
- The term [tex]\(\sqrt[3]{8}\)[/tex] is a cube root of a constant and does not involve a variable.
- The term [tex]\(\sqrt{16}\)[/tex] is a square root of a constant and does not involve a variable.
- There are no variables under any of the radical terms.
- Therefore, this is not considered a radical equation.
After analyzing all the options, the correct answer is:
[tex]\[ \boxed{x \sqrt{3} + x = \sqrt[3]{2x}} \][/tex]
Hence, the correct option is [tex]\(2\)[/tex].
