Let's evaluate each statement step-by-step:
1. Compute [tex]\(3^{-4}\)[/tex]:
[tex]\[
3^{-4} = \frac{1}{3^4} = \frac{1}{81} \approx 0.012345679012345678
\][/tex]
2. Compute [tex]\(\left(\frac{5}{6}\right)^2\)[/tex]:
[tex]\[
\left(\frac{5}{6}\right)^2 = \frac{25}{36} \approx 0.6944444444444445
\][/tex]
3. Compute [tex]\(\left(\frac{5}{6}\right)^3\)[/tex]:
[tex]\[
\left(\frac{5}{6}\right)^3 = \frac{125}{216} \approx 0.5787037037037038
\][/tex]
Now let's compare the values to check each statement:
Statement (A):
[tex]\[
3^{-4} < \left(\frac{5}{6}\right)^2
\][/tex]
This statement is comparing [tex]\(0.012345679012345678\)[/tex] and [tex]\(0.6944444444444445\)[/tex]. Since [tex]\(0.012345679012345678 < 0.6944444444444445\)[/tex], this statement is true.
Statement (B):
[tex]\[
3^{-4} > \left(\frac{5}{6}\right)^3
\][/tex]
This statement is comparing [tex]\(0.012345679012345678\)[/tex] and [tex]\(0.5787037037037038\)[/tex]. Since [tex]\(0.012345679012345678 < 0.5787037037037038\)[/tex], this statement is false.
Statement (C):
[tex]\[
3^{-4} = \left(\frac{5}{6}\right)^3
\][/tex]
This statement is comparing [tex]\(0.012345679012345678\)[/tex] and [tex]\(0.5787037037037038\)[/tex] once more. Since these two values are not equal, this statement is false.
Hence, the true statement is (A).
