To determine which expressions have a value of [tex]\(\frac{16}{81}\)[/tex], we need to evaluate each expression step-by-step.
1. Evaluate [tex]\(\left(\frac{2}{3}\right)^4\)[/tex]:
[tex]\[
\left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4} = \frac{16}{81}
\][/tex]
2. Evaluate [tex]\(\left(\frac{16}{3}\right)^4\)[/tex]:
[tex]\[
\left(\frac{16}{3}\right)^4 = \frac{16^4}{3^4} = \frac{65536}{81}
\][/tex]
3. Evaluate [tex]\(\left(\frac{4}{81}\right)^2\)[/tex]:
[tex]\[
\left(\frac{4}{81}\right)^2 = \frac{4^2}{81^2} = \frac{16}{6561}
\][/tex]
4. Evaluate [tex]\(\left(\frac{4}{9}\right)^2\)[/tex]:
[tex]\[
\left(\frac{4}{9}\right)^2 = \frac{4^2}{9^2} = \frac{16}{81}
\][/tex]
5. Evaluate [tex]\(\left(\frac{1}{91}\right)^{16}\)[/tex]:
[tex]\[
\left(\frac{1}{91}\right)^{16} = \frac{1^{16}}{91^{16}} = \frac{1}{91^{16}}
\][/tex]
After evaluating each expression, we compare each result to [tex]\(\frac{16}{81}\)[/tex]:
- [tex]\(\left(\frac{2}{3}\right)^4 = \frac{16}{81}\)[/tex] : True
- [tex]\(\left(\frac{16}{3}\right)^4 = \frac{65536}{81}\)[/tex] : False
- [tex]\(\left(\frac{4}{81}\right)^2 = \frac{16}{6561}\)[/tex] : False
- [tex]\(\left(\frac{4}{9}\right)^2 = \frac{16}{81}\)[/tex] : True
- [tex]\(\left(\frac{1}{91}\right)^{16} = \frac{1}{91^{16}}\)[/tex] : False
Therefore, the expressions that are equal to [tex]\(\frac{16}{81}\)[/tex] are:
[tex]\[
\left(\frac{2}{3}\right)^4 \quad \text{and} \quad \left(\frac{4}{9}\right)^2
\][/tex]
Conclusively, the correct expressions are:
[tex]\[
\left(\frac{2}{3}\right)^4 \quad \text{and} \quad \left(\frac{4}{9}\right)^2
\][/tex]
