Let's go through the steps in order to simplify the expression [tex]\(\sqrt[4]{567 x^9 y^{11}}\)[/tex]:
1. Start by expressing the entire expression under the 4th root as raised to the power of [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\left(567 x^9 y^{11}\right)^{\frac{1}{4}}
\][/tex]
2. Apply the [tex]\(\frac{1}{4}\)[/tex] power to each element inside the expression separately:
[tex]\[
(81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
3. Split [tex]\(81\)[/tex] into [tex]\(3^4\)[/tex] and apply the [tex]\(\frac{1}{4}\)[/tex] power accordingly:
[tex]\[
(3^4)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
4. Simplify [tex]\( (3^4)^{\frac{1}{4}} \)[/tex] to [tex]\(3\)[/tex]:
[tex]\[
3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
5. Decompose [tex]\(x^{\frac{9}{4}}\)[/tex] and [tex]\(y^{\frac{11}{4}}\)[/tex] into integer and fractional parts:
[tex]\[
3 \cdot (7^{\frac{1}{4}}) \cdot (x^{2} \cdot x^{\frac{1}{4}}) \cdot (y^{2} \cdot y^{\frac{3}{4}})
\][/tex]
6. Re-group the terms to make the expression clearer:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}})
\][/tex]
7. Combine the simplified terms:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot \sqrt[4]{7 x y^3}
\][/tex]
Putting the steps in sequential order:
1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. (81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
3. (3^4)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
4. (81)^{\frac{1}{4}} \cdot(7)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4}+\frac{1}{4}\right)} \cdot y^{\left(\frac{8}{4}+\frac{3}{4}\right)}
5. 3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}})
6. 3 \cdot x^2 \cdot y^2 \cdot \left( 7 x y^3 \right)^{\frac{1}{4}}
