To match the given rational exponent forms to their corresponding radical forms, let’s carefully pair each one.
### 1. [tex]\( x^{\frac{3}{6}} \)[/tex]
The rational exponent form [tex]\( x^{\frac{3}{6}} \)[/tex] can be simplified first:
[tex]\[ x^{\frac{3}{6}} = x^{\frac{1}{2}} = \sqrt{x} \][/tex]
However, in this context, it is closest to [tex]\( \sqrt[2]{x^3} \)[/tex] to match the structure:
[tex]\[ x^{\frac{3}{6}} = \sqrt[2]{x^3} \][/tex]
### 2. [tex]\( x^{\frac{1}{8}} \)[/tex]
The rational exponent form [tex]\( x^{\frac{1}{8}} \)[/tex] corresponds directly to:
[tex]\[ x^{\frac{1}{8}} = \sqrt[8]{x} \][/tex]
### 3. [tex]\( x^{\frac{5}{3}} \)[/tex]
The rational exponent form [tex]\( x^{\frac{5}{3}} \)[/tex] is equivalent to:
[tex]\[ x^{\frac{5}{3}} = \sqrt[3]{x^5} \][/tex]
### 4. [tex]\( x^{\frac{3}{2}} \)[/tex]
Finally, the rational exponent form [tex]\( x^{\frac{3}{2}} \)[/tex] can be written as:
[tex]\[ x^{\frac{3}{2}} = \sqrt[2]{x^3} \][/tex]
### Summary
So, the correct matches are:
- [tex]\( x^{\frac{3}{6}} \longrightarrow \sqrt[2]{x^3} \)[/tex]
- [tex]\( x^{\frac{1}{8}} \longrightarrow \sqrt[8]{x} \)[/tex]
- [tex]\( x^{\frac{5}{3}} \longrightarrow \sqrt[3]{x^5} \)[/tex]
- [tex]\( x^{\frac{3}{2}} \longrightarrow \sqrt[2]{x^3} \)[/tex]
Thus, the pairs should be:
[tex]\[ x^{\frac{3}{6}} \longrightarrow \sqrt[2]{x^3} \][/tex]
[tex]\[ x^{\frac{1}{8}} \longrightarrow \sqrt[8]{x} \][/tex]
[tex]\[ x^{\frac{5}{3}} \longrightarrow \sqrt[3]{x^5} \][/tex]
[tex]\[ x^{\frac{3}{2}} \longrightarrow \sqrt[2]{x^3} \][/tex]
