Let's solve the equation step-by-step:
Given equation:
[tex]\[
\left(\frac{16}{81}\right)^{\frac{5}{4}-\frac{3}{4}} = \left(\frac{2}{3}\right)^x
\][/tex]
First, simplify the exponent on the left-hand side:
[tex]\[
\frac{5}{4} - \frac{3}{4} = \frac{2}{4} = \frac{1}{2}
\][/tex]
So the equation becomes:
[tex]\[
\left(\frac{16}{81}\right)^{\frac{1}{2}} = \left(\frac{2}{3}\right)^x
\][/tex]
Next, simplify [tex]\(\left(\frac{16}{81}\right)^{\frac{1}{2}}\)[/tex]. This is equivalent to finding the square root of [tex]\(\frac{16}{81}\)[/tex]:
[tex]\[
\left(\frac{16}{81}\right)^{\frac{1}{2}} = \sqrt{\frac{16}{81}} = \frac{\sqrt{16}}{\sqrt{81}} = \frac{4}{9}
\][/tex]
Now, rewrite the equation using the simplified left-hand side:
[tex]\[
\frac{4}{9} = \left(\frac{2}{3}\right)^x
\][/tex]
Notice that [tex]\(\frac{4}{9}\)[/tex] can be written as [tex]\(\left(\frac{2}{3}\right)^2\)[/tex]:
[tex]\[
\frac{4}{9} = \left(\frac{2}{3}\right)^2
\][/tex]
So we have:
[tex]\[
\left(\frac{2}{3}\right)^2 = \left(\frac{2}{3}\right)^x
\][/tex]
Since the bases are the same, the exponents must be equal:
[tex]\[
2 = x
\][/tex]
Now, we need to find the value of [tex]\(3x + 4\)[/tex]:
[tex]\[
3x + 4 = 3(2) + 4 = 6 + 4 = 10
\][/tex]
Therefore, the value of [tex]\(3x + 4\)[/tex] is:
[tex]\[
\boxed{10}
\][/tex]
So the correct answer is (2) 10.
