To simplify the expression [tex]\(\sqrt[4]{567 x^9 y^{11}}\)[/tex], we will use properties of rational exponents step-by-step. Here is the correct order of the simplification steps:
1. Rewrite the given expression using rational exponents:
[tex]\[
\left(567 x^9 y^{11}\right)^{\frac{1}{4}}
\][/tex]
2. Factor the expression inside the parenthesis. Since [tex]\(567 = 81 \cdot 7\)[/tex] and [tex]\(81 = 3^4\)[/tex], we get:
[tex]\[
(3^4 \cdot 7 \cdot x^9 \cdot y^{11})^{\frac{1}{4}}
\][/tex]
3. Apply the exponentiation rule to each factor inside the parenthesis:
[tex]\[
\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
4. Simplify [tex]\(\left(3^4\right)^{\frac{1}{4}}\)[/tex] to 3 and leave the others as they are:
[tex]\[
3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
5. Rewrite the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] by breaking them down:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}
\][/tex]
6. Combine the terms having the same [tex]\( \frac{1}{4} \)[/tex] exponent:
[tex]\[
3 x^2 y^2 \left( 7 x y^3 \right)^{\frac{1}{4}}
\][/tex]
7. Rewrite the expression using radical notation:
[tex]\[
3 x^2 y^2 \sqrt[4]{7 x y^3}
\][/tex]
Therefore, the ordered steps are as follows:
1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. [tex]\((3^4 \cdot 7 \cdot x^9 \cdot y^{11})^{\frac{1}{4}}\)[/tex]
3. [tex]\(\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}\)[/tex]
4. [tex]\(3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}\)[/tex]
5. [tex]\(3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right)\)[/tex]
6. [tex]\(3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}}\)[/tex]
7. [tex]\(3 x^2 y^2 \sqrt[4]{7 x y^3}\)[/tex]
Each step builds on the previous one to progressively simplify the initial expression.
