To determine the radical expression of [tex]\(a^{\frac{4}{9}}\)[/tex], let's start by understanding what it means to express a term with a fractional exponent in radical form.
The general rule for converting an expression with a fractional exponent [tex]\(a^{\frac{m}{n}}\)[/tex] to a radical form is given by:
[tex]\[
a^{\frac{m}{n}} = \sqrt[n]{a^m}
\][/tex]
where [tex]\(m\)[/tex] is the numerator and [tex]\(n\)[/tex] is the denominator of the fraction.
Given the expression [tex]\(a^{\frac{4}{9}}\)[/tex]:
- [tex]\(m = 4\)[/tex]
- [tex]\(n = 9\)[/tex]
Using the above rule, we can rewrite [tex]\(a^{\frac{4}{9}}\)[/tex] as:
[tex]\[
a^{\frac{4}{9}} = \sqrt[9]{a^4}
\][/tex]
Now, let’s match this radical expression with the given options:
1. [tex]\(4a^9\)[/tex]
2. [tex]\(9a^4\)[/tex]
3. [tex]\(\sqrt[4]{a^9}\)[/tex]
4. [tex]\(\sqrt[9]{a^4}\)[/tex]
Examining these options, the correct radical form of [tex]\(a^{\frac{4}{9}}\)[/tex] is clearly:
[tex]\(\sqrt[9]{a^4}\)[/tex] (option 4).
Thus, the correct answer is:
[tex]\(\boxed{4}\)[/tex]
