To solve the equation [tex]\(\left(\frac{729}{8x}\right)^{\frac{1}{3}} = \frac{9}{4}\)[/tex], follow these steps:
1. Isolate the fractional exponent:
[tex]\[
\left(\frac{729}{8x}\right)^{\frac{1}{3}} = \frac{9}{4}
\][/tex]
2. Eliminate the cube root by cubing both sides of the equation:
[tex]\[
\left[\left(\frac{729}{8x}\right)^{\frac{1}{3}}\right]^3 = \left(\frac{9}{4}\right)^3
\][/tex]
Simplifying the left side:
[tex]\[
\frac{729}{8x} = \left(\frac{9}{4}\right)^3
\][/tex]
3. Calculate [tex]\(\left(\frac{9}{4}\right)^3\)[/tex]:
[tex]\[
\left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3} = \frac{729}{64}
\][/tex]
4. Set up the equation with the simplified right-hand side expression:
[tex]\[
\frac{729}{8x} = \frac{729}{64}
\][/tex]
5. To solve for [tex]\(x\)[/tex], equate the two fractions:
[tex]\[
\frac{729}{8x} = \frac{729}{64}
\][/tex]
6. Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
729 \cdot 64 = 729 \cdot 8x
\][/tex]
7. Simplify the equation:
Divide both sides by 729:
[tex]\[
64 = 8x
\][/tex]
8. Isolate [tex]\(x\)[/tex] by dividing both sides by 8:
[tex]\[
x = \frac{64}{8}
\][/tex]
9. Simplify the division:
[tex]\[
x = 8
\][/tex]
Thus, the value of [tex]\(x\)[/tex] is [tex]\(\boxed{8}\)[/tex].
