To simplify the expression [tex]\(\left(\frac{81 x^{\frac{2}{3}}}{256 x^{\frac{11}{4}} y^{\frac{2}{3}}}\right)^{\frac{3}{4}}\)[/tex], let's proceed step-by-step:
1. Simplify the fraction inside the parentheses:
[tex]\[
\frac{81 x^{\frac{2}{3}}}{256 x^{\frac{11}{4}} y^{\frac{2}{3}}}
\][/tex]
First, handle the [tex]\(x\)[/tex]-terms by combining the exponents. The combined exponent for [tex]\(x\)[/tex] can be found by subtracting the exponent in the denominator from the exponent in the numerator:
[tex]\[
x^{\frac{2}{3}} \div x^{\frac{11}{4}} = x^{\left(\frac{2}{3} - \frac{11}{4}\right)}
\][/tex]
To subtract these fractions, we need to have a common denominator. The least common multiple of 3 and 4 is 12:
[tex]\[
\frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}
\][/tex]
[tex]\[
\frac{11}{4} = \frac{11 \cdot 3}{4 \cdot 3} = \frac{33}{12}
\][/tex]
[tex]\[
\frac{2}{3} - \frac{11}{4} = \frac{8}{12} - \frac{33}{12} = \frac{8 - 33}{12} = \frac{-25}{12}
\][/tex]
So, the exponent for [tex]\(x\)[/tex]-terms becomes [tex]\(x^{-\frac{25}{12}}\)[/tex].
We now rewrite the expression inside the parentheses as:
[tex]\[
\frac{81 x^{-\frac{25}{12}}}{256 y^{\frac{2}{3}}}
\][/tex]
2. Apply the exponent [tex]\(\frac{3}{4}\)[/tex] to the entire expression:
[tex]\[
\left( \frac{81 x^{-\frac{25}{12}}}{256 y^{\frac{2}{3}}} \right)^{\frac{3}{4}}
\][/tex]
Distribute the [tex]\(\frac{3}{4}\)[/tex] exponent to each part of the fraction:
[tex]\[
\left( \frac{81}{256} \right)^{\frac{3}{4}} \left( x^{-\frac{25}{12}} \right)^{\frac{3}{4}} \left( y^{-\frac{2}{3}} \right)^{\frac{3}{4}}
\][/tex]
3. Simplify each term separately:
- For the constant term:
[tex]\[
\left( \frac{81}{256} \right)^{\frac{3}{4}}
\][/tex]
Recognize that 81 and 256 are perfect powers:
[tex]\[
81 = 3^4 \quad \text{and} \quad 256 = 4^4
\][/tex]
Therefore,
[tex]\[
\left( \frac{81}{256} \right)^{\frac{3}{4}} = \left( \frac{3^4}{4^4} \right)^{\frac{3}{4}} = \left( \frac{3}{4} \right)^3 = \frac{27}{64}
\][/tex]
- For the [tex]\(x\)[/tex]-term:
[tex]\[
\left( x^{-\frac{25}{12}} \right)^{\frac{3}{4}} = x^{-\frac{25}{12} \cdot \frac{3}{4}} = x^{-\frac{75}{48}}
\][/tex]
- For the [tex]\(y\)[/tex]-term:
[tex]\[
\left( y^{\frac{2}{3}} \right)^{\frac{3}{4}} = y^{\frac{2}{3} \cdot \frac{3}{4}} = y^{\frac{1}{2}}
\][/tex]
4. Combine all simplified terms:
[tex]\[
\frac{27}{64} x^{-\frac{75}{48}} y^{-\frac{1}{2}}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{\frac{27 x^{-\frac{75}{48}} y^{-\frac{1}{2}}}{64}}
\][/tex]
