To evaluate the expression [tex]\(\left(\frac{16}{81}\right)^{\frac{1}{2}}\)[/tex], we follow these steps:
1. Understand the exponent [tex]\(\frac{1}{2}\)[/tex]:
- The exponent [tex]\(\frac{1}{2}\)[/tex] signifies the square root of the number.
- Therefore, [tex]\(\left(\frac{16}{81}\right)^{\frac{1}{2}}\)[/tex] is equivalent to [tex]\(\sqrt{\frac{16}{81}}\)[/tex].
2. Simplify the fraction inside the square root:
- The square root of a fraction [tex]\(\frac{a}{b}\)[/tex] can be expressed as [tex]\(\frac{\sqrt{a}}{\sqrt{b}}\)[/tex].
- Applying this property, we have:
[tex]\[
\sqrt{\frac{16}{81}} = \frac{\sqrt{16}}{\sqrt{81}}
\][/tex]
3. Find the square root of the numerator and the denominator:
- The square root of 16 is 4 because [tex]\(4^2 = 16\)[/tex].
- The square root of 81 is 9 because [tex]\(9^2 = 81\)[/tex].
4. Divide the square roots:
- Using the values found, we get:
[tex]\[
\frac{\sqrt{16}}{\sqrt{81}} = \frac{4}{9}
\][/tex]
5. Final answer:
- Therefore, [tex]\(\left(\frac{16}{81}\right)^{\frac{1}{2}} = \frac{4}{9}\)[/tex].
Given the result [tex]\(\frac{4}{9}\)[/tex], the correct answer is:
[tex]\[ \boxed{\frac{4}{9}} \][/tex]
This matches the numerical result obtained, confirming that the solution is correct.
